Angle trisector, as validated to perform accurately over a wide range of device settings by a novel geometric forming process; also capable of portraying finite lengths that only could be approximated by means of otherwise applying a compass and straightedge to a given length of unity; that furthermore functions as a level whose inherent geometry could be adapted for many other uses such as being incorporated into the design of a hydraulic car lift.

ABSTRACT

A newly proposed articulating invention, each of whose four constituent embodiments is designed to trisect any of a multitude of suitably described angles by means of becoming properly set to its designated magnitude; thus automatically portraying a motion related solution for the trisection of an angle that discloses complete routing details of a pathway that leads from such designated magnitude all the way back to its trisector; thereby discerning the whereabouts of certain intersection points which evade detection when attempting to otherwise locate them by means of applying only a straightedge and compass to an angle of such designated magnitude; furthermore projecting finite lengths of any trisector that bears cubic irrational trigonometric properties, being those that cannot be duplicated, but only approximated, when applying a straightedge and compass to a given length of unity; and being of a unique design that could be adapted to function as a level.

CROSS-REFERENCE TO RELATED APPLICATIONS

This nonprovisional patent application is a continuation applicationwhich claims the benefit of U.S. application Ser. No. 15/889,241, filedFeb. 6, 2018, now pending, and hereby is incorporated by reference.

BACKGROUND OF THE INVENTION

By now, it most certainly should be recognized that a singular angle ofvirtually any designated size cannot be divided into three parts whenacted upon only by a straightedge and a compass.

Such protracted problem is considered to be so famous that herein itformally shall be referred to as the classical problem of the trisectionof an angle!

Unfortunately, many other descriptions of such problem also exist, eachconflicting in some rather subtle manner, but nevertheless havingprofound effect upon its interpretation. Leading examples of suchdifferences are presented below where it is found that in certain casesthroughout the twentieth century such problem of the trisection of anangle:

is not specified, but its solution nevertheless is posed;

is considered to involve the use of a ruler; and

is algebraically, rather than geometrically resolved.

A looming trisection mystery, steeped in controversy for millennia,persists right up until present day; merely because it never wasexamined from the proper perspective!

In order to crack such conundrum, not only do correct questions need toasked, but proper answers also need to be supplied.

In this regard, the very first of four fundamental questions is about tobe posed.

The first riddle is: How can the classical problem of the trisection ofan angle actually be solved?

The answer is: It cannot!

Granted, it is commonly known that such response has been hotlycontested by many pundits over the years.

But, to its credit, after bearing the brunt of a constant barrage ofbrutal assaults by noted protagonists, such contention endures; wherebyit appears that to date no solution for the classical problem of thetrisection of an angle ever has been solicited that has withstood thetest of time!

Because it is far more difficult to prove an impossibility thansomething that is thought to be possible, fewer proofs exist whichattempt to validate that the classical problem of the trisection of anangle truly is unsolvable!

To this effect, page 246 appearing in A Dictionary of Mathematics by T.A. Millington, Barnes & Noble; 1966 stated, “The classical problem oftrisecting an angle by Euclidean constructions (use of straight linesand circles only) was proved to be insoluble by Wantzel in 1847”.

Below, my express intent is to acknowledge Wantzel's outright conclusionas being absolutely correct in such regard, in hopes of therebyidentifying some distinct difficulty which prevails that otherwiseprevents the classical problem of the trisection of an angle from beingsolved!

This is to be accomplished by means of delving into what before wereconsidered to be impenetrable depths of a great enigma in order tofinally unravel the very mystery that today engulfs trisection.

Such trisection enigma specifically shall state: If the classicalproblem of the trisection of an angle truly cannot be solved, thengeometry, itself, must be the culprit; thereby being imperfect!

This author well understands that if it were learned that geometryactually is limited, marred, or otherwise flawed in some strange,presently unknown fashion, then the very perception of such form ofmathematics inevitably would become altered, even to the point where itpossibly might become tarnished forevermore!

The second riddle is: What specific, well known thermodynamicslimitation now should be considered to furthermore apply to geometry?

The answer is: Irreversibility!

A commentary, as presented below, should settle any doubts, or quell anylingering suspicions, that the classical problem of the trisection of anangle might become solved.

A fair number of rather elaborate proofs that have gained notoriety overthe years have done so by means of incorrectly claiming that theclassical problem of the trisection of an angle actually can be solved.

Such proofs either are quite faulty in their logic, or else rely upon abasic deception; whereby their geometric solutions, although complyingwith a primary requirement of applying only a straightedge and compass,achieve trisection instead by having them act upon other than a singularangle of virtually any designated size!

In such manner, geometric solutions actually do achieve trisection, butonly to their detriment by means of violating a remaining requirementthat the classical problem of the trisection of an angle furthermoreimposes!

As such, for any and all geometric solutions which might becomespecified in the near future claiming to succeed at trisecting an angle,perhaps it is best to offer the rather simple-minded reservation that inorder to solve the classical problem of the trisection of an angle, allof its imposed requirements explicitly must be complied with!

In other words, the very reason why any of such actual geometricsolutions should not be misinterpreted as a solution for the classicalproblem of the trisection of an angle is because they violate the secondof its requirements, fundamentally specifying that geometric activitymust proceed only from a singular angle of virtually any designatedsize.

Instead of complying with such second requirement, geometric solutionsotherwise input extraneous information into such classical problem,thereby serving to corrupt it!

In particular, extraneous information consists of that which is notgermane to such classical problem. Hence it consists of any geometricinformation other than that which can be derived by applying only astraightedge and compass exclusively to a singular angle of designatedmagnitude which is intended to be trisected.

Moreover, extraneous information is considered to be relevant wheneverit independently can be acted upon solely by a straightedge and compassin order to pose an actual solution for some corrupted version of theclassical problem of the trisection of an angle; thereby solving anentirely different problem!

Hence, it becomes rather obvious that such very controversy undeniablyhas been fueled over the years by an unrelenting confusion which hasbeen stirred over nothing more than various cases of mistaken identity!

A rather elementary example of how to administer such overall theory ispresented directly below; whereby for the unsolvable classical problemof the trisection of an angle of 135° designated magnitude, it isdesired to specify relevant information that enables a geometricsolution to be obtained.

For such example, the approach to obtain such relevant information isgiven as follows:

extraneous information quite readily can be determined simply bydividing such 135° angle by a factor of three. Such resulting 45 sizequalifies as extraneous information because it clearly cannot be derivedin any way by means of applying only a compass and straightedge to such135° designated angle; and

such calculated 450 magnitude furthermore is relevant because it can bedrawn solely by application of a straightedge and compass as one of thediagonals in a square of virtually any size, thereby posing a geometricsolution for such corrupted classical trisection problem.

Such types of mishap, assuming the form of mistaken identifies, veryeasily can be avoided whenever any geometric solution becomes posed inthe future that falsely alleges that the classical problem of thetrisection of an angle actually can be solved.

Quite simply, such approach consists of determining whether a posedgeometric solution is linked to a formally specified problem. If acorresponding problem becomes located, it then should be examined toverify that it:

is entirely consistent with how such classical problem of the trisectionof an angle is specified above, in which case such geometric solutiontherefore must be in error; or

incorporates extraneous information that is relevant, thereby otherwisesolving some corrupted version of such classical trisection probleminstead.

On the other hand, if it is found that such geometric solution is notassociated to any formally specified problem, this means that it cannotbe ascertained to any certainty what such posed geometric solution evenapplies to!

Regrettably, such type mishap is well documented, thereby being known totruly have occurred in the past!

As such, for any literature that claims to have solved the classicalproblem of the trisection of an angle without indicating exactly whatparticular issue it has remedied, it might be a good idea to examine:

whether such erroneous claim was met by a great fanfare that creditedits author for an outstanding discovery.

whether a geometric solution for some corrupted version of the classicalproblem of the trisection of an angle was provided instead, withoutexpressing the exact nature of extraneous information which suchsolution was based upon; or

whether some totally unrelated type of trisection solution wasidentified due to the discovery of some invention; thereby only servingto expand upon the overall scope of aforementioned trisection matters ascited in connection with such attendant trisection mystery.

The third riddle is: What other types of trisection solutions, besidesgeometric solutions, are there?

The answer is: Motion related solutions!

These consist of various events, as opposed to activities where singulargeometric patterns otherwise become drawn, whereby an invention becomesset to a singular angle of virtually any designated size and itsassociated trisector automatically becomes portrayed forthwith.

Such type of invention cannot solve the classical problem of thetrisection of an angle because its features are quite different from amere straightedge and compass that instead individually must be appliedto a single piece of paper without otherwise violating any imposedrequirements. Hence, any trisection portrayal of this nature clearlywould qualify as a corrupted motion related solution of the classicalproblem of the trisection of an angle.

The fourth riddle is: Can an iteration process of successive angularbisections, as presented in complete detail later, which clearlycomplies with all of the requirements imposed by the classical problemof the trisection of an angle actually solve it?

The answer again emphatically is no!

In particular, this is because:

an infinite number of iterations would have to be performed in order togenerate an exact geometric solution; thereby qualifying as a task thatcould never be fully completed; and

the resolution of the naked eye to distinguish actual drawingseparations would become impaired very shortly after commencing uponsuch iteration process, whereby successive bisectors then would appearto run over themselves, requiring larger arcs to be drawn in order toavail added viewing relief. Eventually, the very straightedge andcompass instruments themselves, along with the paper needed to availsuch precision could not be manufactured due to the massive increases intheir sizes which would be needed to maintain such viability.

In conclusion, it is impossible to solve the classical problem of thetrisection of an angle when explicitly complying with all of itsrequirements!

Important forerunners of trisection, hereinafter listed chronologicallyby the dates of their conception include:

geometric construction, dating all the way back to ancient Babyloniantimes around the year 3000 BC, in which only a straightedge and compassare permitted for use when describing straight lines, angles, andcircular arcs;

intersection points, as established during the same time period, inwhich discrete positions become completely distinguishable wherevereither straight lines, arcs of circles, or straight lines in combinationwith the arcs of circles cross one another. Center points of circlesalso qualify as intersection points because they describe commonlocations where geometrically constructed perpendicular bisectors ofrespective chords of such circles meet;

the Pythagorean Theorem, as developed in or about the year 500 BC, inwhich the square of the length of the hypotenuse of a right triangle isproven to be equal to the sum of the squares of the lengths of its twosides;

conventional Euclidean practice, as established prior to the year 265BC, in which definitions and rules describe the very manner in whichgeometric construction may be administered, and axioms identify certaingeometric relationships which become evident after conducting specificgeometric construction operations. Directly below, two principaldefinitions which further characterize such practice are specified,followed by three distinct examples of its rules:

its most basic rule principally states that a given set of previouslydefined geometric data must be furnished that specifies the locations ofinitial positions from which geometric construction may be launched. Adesignated angle of sixty degree magnitude very well could be expressedas such given data. Moreover, such definition furthermore can apply tolocations that are not entirely distinguishable solely by geometricconstruction. For example, given data might specify a twenty degreeangle; one whose magnitude only could be approximated by means ofgeometric construction;

its cardinal rule essentially stipulates that geometric constructionmust proceed either from a given set of previously defined geometricdata, or from other locations which become distinguishable with respectto such positions by means of applying only a straightedge and compassto them;

at least two points must be specified in order to draw a straight line;

at least two points must be specified in order to draw a circle when oneof those points designates its center point; and

at least three points must be specified in order to draw a circle whennone of those points denotes its center point;

conventional Euclidean means, whose terminology became commonplaceshortly thereafter, in which geometric construction is to be implementedin strict accordance with the specific definitions and rules stipulatedby conventional Euclidean practice;

Euclidean commands, also taking effect during the same time period, inwhich various instructions describe exactly how various straight linesand circular arcs are to be geometrically constructed with respect toidentified positions;

a sequence of Euclidean operations, as conceived during the sametimeframe, in which a specific set of Euclidean commands, enumerated asa series of discrete steps become executed in consecutive order withrespect to a given set of previously defined geometric data in order todistinguish various rendered positions;

a geometric construction pattern, as introduced during that time, inwhich the specific features of a given set of previously definedgeometric data, in combination with any rendered positions which becomeestablished by means of executing an attendant sequence of Euclideanoperations with respect to it, as well as any additional straight linesdrawn between such established locations or circular extensions made tothem become depicted within a single drawing;

an Archimedes proposition, as devised prior to 212 BC, introduced asProposition 8 in the Book of Lemmas, then later translated from Arabicinto Latin in 1661, and finally published in the English language in1897, in which it is stated on page 309 in The Works of Archimedes thatif AB be any chord of a circle whose centre is 0, and if AB be producedto C so that BC is equal to the radius; if further CO meets the circlein D and be produced to meet the circle a second time in E, the arc AEwill be equal to three times the arc BD;

an Archimedes proof for such Proposition 8, as appears on page 310 inThe Works of Archimedes, in which it is stated:

draw the chord EF parallel to AB, and join radius OB, OF;

$\begin{matrix}{{\angle \; {COF}} = {2\mspace{14mu} \angle \; {OEF}}} \\{{= {2\mspace{14mu} \angle \; {BCO}}},{{by}\mspace{14mu} {parallels}},} \\{{= {2\mspace{14mu} \angle \; {BOD}}},{{{{since}\mspace{14mu} {BC}} = {BO}};}}\end{matrix}$

therefore, ∠BOF=3 ∠BOD, so that the arc BF is equal to three times thearc BD; and

hence the arc AE, which is equal to the arc BF, must be equal to threetimes the arc BD;

an Archimedes formulation, as evidenced only on a few sporadic occasionsin the past, as typically accompanied by only partial documentation, inwhich a multitude of distinct Archimedes geometric constructionpatterns, qualifying as such because they conform to all requirementsposed in such Archimedes proposition, become represented upon a singledrawing. Such representation is made possible by strategically placing aGreek letter either within or alongside what later will be shown to bethe given angle of such sole diagram, thereby distinguishing it to be anentire formulation in itself, instead of a mere singular geometricconstruction pattern. Once ascribing a specific designation, such as θor φ at such location, it is to mean that such given angle is allowed tovary infinitesimally in size over some prescribed range of values. Bymeans of administering a specific sequence of Euclidean operations toeach of such given angles—one whose commands account for all of theindividual requirements posed in such aforementioned Archimedesproposition, all of the varying shapes which emerge thereby must qualifyas legitimate Archimedes geometric construction patterns; and

an Al-Mahani categorization, as derived prior to the year 900 AD, inwhich square root quantities become classified as quadratic irrationalnumbers, thereby distinguishing them apart from rational numbers.

Over the years, both mathematicians, as well as inventors alike havebeen somewhat awed by the spectacle of an incredibly perplexingtrisection mystery whose vital secrets evidently have escaped detection.

Nevertheless, both parties are acutely aware that the diminution of anangle to one-third its actual size, although being truly indicative oftrisection, cannot be obtained simply by reversing the sequence ofEuclidean operations which governs the geometric construction of apattern that complies with all of the requirements imposed by suchArchimedes proposition.

Rabid speculation concerning such unexplainable dichotomy eventuallygave rise to contrasting interpretations, reflective of the particularleanings of various involved personages, outlined as follows:

on the one hand, mathematicians even today remain splintered over how toexplain trisection in terms of conventional Euclidean practice, wherebytheir discordant positions concerning such geometrical matters areexpressed as follows:

one traditionalist camp contends that trisection solely via conventionalEuclidean means is entirely unsolvable; whereby it becomes utterlyimpossible to divide an angle into three parts solely by means ofapplying only a straightedge and compass to it; whereas

another non-patronizing faction instead advocates that certain angles,in fact, can be trisected solely via straightedge and compass; whereas

on the other hand, inventors were the first members in society todemonstrate that trisection could be achieved by imparting certainprescribed motions.

The latter of such two geometrical claims, essentially alleging thatcertain angles can be trisected solely via straightedge and compassunfortunately has managed to gain widespread notoriety throughout theworld today, thereby flourishing in the form of independent airings by acoterie of indulging sources and journalists, as well as beingsupplemented by a rampant proliferation of publications byself-proclaimed mathematical experts who already have accepted suchpremise as being generally established fact.

Be that as it may, nothing could be further from the truth!

This becomes evident by paraphrasing such latter stated mathematicalclaim into its only possible correct interpretation. This isaccomplished by means of inserting the following bracketed italicizedwords to its initial substance, thereby asserting that certain angles,in fact, can be trisected solely via straightedge and compass [so longas the very magnitudes of their respective trisecting angles becomedisclosed beforehand]—thereby essentially solving a corrupted version ofthe classical problem of the trisection of an angle.

Such above stated clarification achieves two specific objectives,itemized as follows:

it completely eliminates the potential for misconstruing such latterstated mathematical claim to mean that in certain cases, trisectionsolely via conventional Euclidean means is entirely possible Obviously,such faulty presumption might be harmful because it very easily could beconsidered to refute, or contradict such former factually correct claim,as asserted above by such aforementioned traditionalist camp ofmathematicians; and

it evidences that such latter mathematical clause really has nothing atall to do with trisection solely via conventional Euclidean means;thereby removing the intended stigma out of such statement entirely. Assuch, its true interpretation with such bracketed input included therebybecomes reduced to the rather insignificant equivalent meaning that allangles which can be described solely via straightedge and compassfurthermore constitute respective trisecting angles of other angleswhose respective magnitudes amount to exactly three times their size.Such result is of little consequence too considering that it furthermorecan be broken down into the mere definition of a trisector, in concertwith the understanding that certain angles effectively can be reproducedsolely via straightedge and compass once having knowledge of theirtrigonometric properties.

It should be emphasized that the only way to input outside informationwhich is relevant to the classical problem of the trisection of an angleis by means of violating the cardinal rule of conventional Euclideanpractice!

By definition, that is because such input must consist of data thatcannot be derived by means of launching geometric constructionoperations exclusively from a singular angle of designated magnitudewhich is intended to be trisected. Hence, such corrupting input mustconstitute that which cannot be distinguished by conventional Euclideanmeans.

As such, it is concluded that the classical problem of the trisection ofan angle cannot be solved by conventional Euclidean practice; therebyexposing its very limitations!

Furthermore, calculating the size of a trisector merely by dividing thedesignated magnitude of an angle that is intended to be trisected by afactor of three is not permitted because such action cannot beduplicated by a geometric construction process which is governed byEuclidean rules.

Below, a composite history describing the very first importanttrisecting events recorded in the English speaking language is afforded;whereby trisection was observed to occur on three separate occasions asunique articulating mechanisms became invented.

During such first of such documented incidents, having taken placesometime during the late 1870's, Alfred Kempe discovered that so-calledanti-parallelograms could be used for purposes of regulating motion!

One of such Kempe masterful designs, truly considered to be capable ofperforming trisection, is depicted as prior art in FIG. 1A.

Therein, an overall formation which could have been fashioned to haveeither rounded or pointed corners is not displayed. Only thelongitudinal centerlines of its eight linkages and radial centerlines ofits eight interconnecting pivot pins are depicted instead because onlythese portions of such device govern trisection!

Such Kempe prior art features a basic fan array whose movement iscontrolled by three independent anti-parallelograms; the completebreakdowns of which are described as follows:

its fan array portion consists of four separate linkages, modeled asstraight lines BA, BD, BE, and BC in FIG. 1A; all of which are hingedtogether by an interconnecting pivot pin whose shank passes through anend portion of each. In particular, the longitudinal centerlines of suchfour basic fan array linkages overlap one another along the radialcenterline of such interconnecting pivot pin, as illustrated by a verysmall circle drawn located at their juncture about axis B; and

its control section features anti-parallelograms BFGH, BGJI, and BJLK,so denoted by identifying their diagonal compositions, rather than byenumerating their respective corners in consecutive order.

Such control section serves to maintain the three angles interposedbetween adjacent longitudinal centerlines of the four linkages, wherethey more particularly radiate about the hub of such basic fan arrayportion, at a constant magnitude; even as such sizes vary during deviceflexure.

Accordingly, such three interposed angles relate to one anotheraccording to the algebraically expressed equality ∠ABD=∠DBE=∠EBC=θ.

Whereas all three subtended angles in combination constitute angle ABC,its magnitude must amount to 3θ.

In order to trisect angle ABC, such Kempe device first must be set to adesignated magnitude. In this particular case, the size of such settingis shown to be that of angle ABC, as it actually appears in FIG. 1A.Considering that a varying magnitude of such angle ABC might have beenselected instead, such drawing would have had to assume an entirelydifferent overall shape in order to compensate for such change.

In effect, every time such Kempe device becomes set to a differentpreselected magnitude, each of such three subtended angles, nonetheless,automatically portray its actual trisector; whereby the very process ofportraying an angle whose magnitude amounts to exactly one-third thesize of an angle of designated magnitude is indicative of trisection!

In total, such Kempe device is comprised of eight linkages whoselongitudinal centerlines are modeled as straight lines in FIG. 1A, inaddition to eight interconnecting pivot pins whose radial centerlinesare denoted by very small circles therein.

Extensions made to anti-parallelogram linkage members BF and BL thereinenable easier access for flexing device arm BC with respect to linkagemember BA during articulation; whereas extensions made toanti-parallelogram linkages members BG and BJ enable a better viewing oftrisecting members BD and BE.

All told, FIG. 1A depicts longitudinal centerlines of linkages andradial centerlines of interconnecting pivot pins which collectivelyconstitute such Kempe mechanism.

Of note, various other trisecting devices exist which also are contrivedof a four linkage and interconnecting pivot pin arrangement which areconsidered to conform to that of such basic fan array, as described indetail above. However, they each apply control mechanisms that areentirely different than the anti-parallelogram linkage arrangements, aspreviously developed by Mr. Kempe.

By means of classifying mechanisms such as these into a singularcategory, it becomes possible to validate the striking geometricalresemblance which exists between them. Once grouped together, theiruniqueness thereby can be substantiated by means of describing how theircontrol mechanisms differ from each other.

Such categorization is necessary for the very same reason that itpreviously has been applied in biology; namely, to suitably characterizethe very diversity which exists between living things that exhibitcommon physical traits.

Likewise, by means of comparing trisecting mechanisms which exhibitsimilar geometries, proper conclusions can be drawn concerning both howand why they relate to one another, as well as how they fundamentallydiffer!

What should bond trisecting mechanisms together is a common geometrywhich they all share.

For example, the geometry of such basic fan array design is perhaps thesimplest in all of mathematics to comprehend because it simply consistsof a given angle that becomes duplicated twice, such that all threeangles become grouped together at a common vertex in order to eventuallygeometrically construct a rendered angle whose magnitude amounts toexactly three times the size of such given angle.

A singular geometric construction pattern, as represented in FIG. 1A,furthermore could qualify as an entire formulation because the sequenceof Euclidean operations from which such distinctive geometricconstruction pattern stems furthermore could be applied to virtually anysized given angle, as algebraically denoted therein by the Greek letterθ; thereby governing the various positions which the longitudinalcenterlines of linkages and radial centerlines of interconnecting pinswhich collectively constitute such Kempe device thereby would assume asit becomes articulated.

In support of such logic, it is recommended that any articulatingtrisecting mechanism which exhibits distinctive fan shape featuresshould be classified as a CATEGORY I type device. Its complete inventoryshall consist of:

CATEGORY I, sub-classification A articulating trisection devices whichfeature four linkages of equal length, hinged together by aninterconnecting pivot pin that is passed through one end portion ofeach, thereby collectively constituting the array of a fan whose twoinner linkages become regulated in some fashion so that theirlongitudinal centerlines divide the angle subtended by the longitudinalcenterlines of its two outer linkages into three equal portionsthroughout device flexure; and

CATEGORY I, sub-classification B articulating trisection devices whichfeature three linkages of equal length, hinged together by aninterconnecting pivot pin that is passed through one end portion ofeach, either of whose outer linkages instead could be represented by astraight line that is impressed upon a piece of paper or board, therebycollectively constituting the array of a fan whose single inner linkagebecomes regulated in some fashion so that its longitudinal centerlinetrisects the angle subtended by the longitudinal centerlines of its twoouter linkages throughout device flexure.

With particular regard to FIG. 1A, each of the distinct Euclideancommands required to suitably locate straight line BC with respect togiven angle ABD therein, solely by Euclidean means, is specified below,thereby together comprising the complete twenty-one steps of itssequence of Euclidean operations:

step 1—given angle ABD, of arbitrarily selected magnitude θ, is drawnsuch that its side BA is constructed to be of equal length to its otherside BD;

step 2—point F is arbitrarily selected somewhere along side BA of givenangle ABD;

step 3—a circle is drawn about vertex B whose radius is arbitrarilyselected to be less than length BF, whereby a portion of itscircumference becomes designated as the FIRST CIRCULAR ARC in FIG. 1A;

step 4—The intersection between such FIRST CIRCULAR ARC and side BD ofgiven angle ABD becomes designated as point G;

step 5—a second circle is drawn about point F whose radius is set equalin length to straight line segment BG, a portion of whose circumferenceis designated as the SECOND CIRCULAR ARC in FIG. 1A;

step 6—a third circle is drawn about point G whose radius is set equalin length to straight line segment BF, a portion of whose circumferenceis designated as the THIRD CIRCULAR ARC in FIG. 1A;

step 7—the intersection point between such SECOND CIRCULAR ARC and THIRDCIRCULAR ARC is designated as point H;

step 8—diagonal GH and side segment FH of anti-parallelogram BFGH aredrawn in order to complete its geometry;

step 9—an angle whose magnitude is equal to that of angle FBH isgeometrically constructed with its vertex placed at point B such thatits counterclockwise side becomes aligned along straight line BD,whereby the intersection of its clockwise side with diagonal GH becomesdesignated as point I;

step 10—a fourth circle is drawn about point I, whose radius is setequal in length to line segment BG, a portion of whose circumferencebecomes designated as the FOURTH CIRCULAR ARC in FIG. 1A;

step 11—a fifth circle is drawn about point B, whose radius is set equalin length to line segment IG, a portion of whose circumference isdesignated as the FIFTH CIRCULAR ARC in FIG. 1A;

step 12—the intersection point between such FOURTH CIRCULAR ARC andFIFTH CIRCULAR ARC becomes designated as point J;

step 13—straight line diagonal IJ and side segment BJ are drawn in orderto complete an additional anti-parallelogram BGJI;

step 14—straight line BE is geometrically constructed to be equal inlength to side BA of given angle ABD such that it aligns with sidesegment BJ of such additional anti-parallelogram BGJI by means ofpassing through point J, whereby it serves as an extension to it;

step 15—an angle whose magnitude is equal to that of angle GBI isgeometrically constructed with its vertex placed at point B such thatits counterclockwise side becomes aligned along straight line BE wherebythe intersection of its clockwise side with diagonal JI becomesdesignated as point K;

step 16—a sixth circle is drawn about point K, whose radius is set equalin length to line segment BJ, a portion of whose circumference isdesignated as the SIXTH CIRCULAR ARC in FIG. 1A;

step 17—a seventh circle is drawn about vertex B of given angle ABDwhose radius is set equal in length to line segment KJ, a portion ofwhose circumference is designated as the SEVENTH CIRCULAR ARC in FIG.1A;

step 18—the intersection point between such SIXTH CIRCULAR ARC andSEVENTH CIRCULAR ARC is designated as point L;

step 19—straight line diagonal KL and side segment BL are drawn in orderto complete the third and last of such anti-parallelograms, being dulynotated as BJLK;

step 20—straight line BC is geometrically constructed to be equal inlength to side BA of given angle ABD such that it aligns with sidesegment BL of anti-parallelogram BJLK by means of passing through pointL, thereby serving as an extension to it; and

step 21—whereas side BA is constructed to be of equal length to side BDof given angle ABD, and straight lines BE and BC are geometricallyconstructed to be equal in length to such side BA, all four straightlines furthermore represent radii of a circle that all emanate fromcenter point B.

Verification that angle ABC, as depicted in FIG. 1A, is equal to threetimes the size of given angle ABD is provided in the followingtwenty-five step proof:

step 1—by construction, straight line HG is equal in length to straightline BF, and straight line GB is equal in length to straight line FH;

step 2—by identity, straight line BH is equal in length to straight lineHB;

step 3—then, since the three sides represented in triangle HGB are ofequal length to the corresponding sides of triangle BFH, such trianglesmust be congruent to each other;

step 4—by construction, straight line IJ is equal in length to straightline BG, and straight line JB is equal in length to straight line GI;

step 5—by identity, straight line BI is equal in length to straight lineIB;

step 6—then, since the three sides represented in triangle IJB are ofequal length to the corresponding sides of triangle BGI, such trianglesmust be congruent;

step 7—by construction, straight lines KL and LB respectively are equalin length to straight lines BJ and JK, and by identity, straight line BKis equal in length to straight line KB;

step 8—then, since the three sides represented in triangle KLB are ofequal length to the corresponding sides of triangle BJK, such trianglesmust be congruent to each other;

step 9—since triangle BFH is congruent to triangle HGB, each of theircorresponding angles must be of equal magnitudes, such that ∠FBH=∠GHB;

step 10—but, by construction ∠GBI=∠FBH;

step 11—then, by substitution ∠GBI=∠GHB;

step 12—by identity, ∠BGI=∠BGH;

step 13—since ∠GBI=∠GHB, in addition to the fact that ∠BGI=∠BGH,triangle BGI and triangle HGB contain two sets of angles whoserespective magnitudes are of equal values, whereby such triangles mustbe similar to one another;

step 14—because triangle IJB is congruent to a triangle BGI which, inturn, is similar to triangle HGB, triangle IJB also must be similar totriangle HGB;

step 15—whereby angle IBJ must be equal to corresponding angle HBG;

step 16—because triangle BGI is congruent to triangle IJB, each of theircorresponding angles must be of equal magnitudes, such that ∠GBI=∠JIB;

step 17—but, by construction ∠JBK=∠GBI;

step 18—then, by substitution ∠JBK=∠JIB;

step 19—by identity, ∠BJK=∠BJI;

step 20—since ∠JBK=∠JIB, in addition to the fact that ∠BJK=∠BJI,triangle BJK and triangle IJB contain two sets of angles whoserespective magnitudes are of equal values, whereby such triangles mustbe similar to one another;

step 21—because triangle KLB is congruent to a triangle BJK which, inturn, is similar to triangle IJB, triangle KLB also must be similar totriangle IJB;

step 22—whereby angle KBL must be equal to corresponding angle IBJ;

step 23—since the whole is equal to the sum of its parts,

∠HBA+ABD=∠HBD

∠ABD=∠HBD−∠HBA

∠ABD=∠HBD−∠HBF

∠ABD=∠HBD−∠FBH;

step 24—whereby the following expression is obtained by substitutingrelevant previously determined equalities and identities ∠HBD=∠HBG,∠IBJ=∠IBE, ∠KBJ=∠KBE, ∠KBL=∠KBC, and ∠IBG=∠IBD, and by reversing theorder of the three letter designators of certain angles withoutinfluencing the values of their respective magnitudes, such that:

$\begin{matrix}{{\angle \; {ABD}} = {{\angle \; {HBD}} - {\angle \; {FBH}}}} \\{= {{\angle \; {HBG}} - {\angle \; {FBH}}}} \\{= {{\angle \; {IBJ}} - {\angle \; {GBI}}}} \\{= {{\angle \; {IBE}} - {\angle \; {IBG}}}} \\{= {{{\angle \; {IBE}} - {\angle \; {IBD}}} = {\angle \; {DBE}}}} \\{= {{\angle \; {IBJ}} - {\angle \; {GBI}}}} \\{= {{\angle \; {KBL}} - {\angle \; {JBK}}}} \\{= {{\angle \; {KBC}} - {\angle \; {KBJ}}}} \\{{= {{{\angle \; {KBC}} - {\angle \; {KBE}}} = {\angle \; {EBC}}}};}\end{matrix}\quad$

and

step 25—since angle ABC is comprised of angle ABD, angle DBE, and angleEBC, the following expression demonstrates that angle ABC amounts toexactly three times the size of given angle ABD:

$\begin{matrix}{{\angle \; {ABC}} = {{\angle \; {ABD}} + {\angle \; {DBE}} + {\angle \; {EBC}}}} \\{= {{\angle \; {ABD}} + {\angle \; {ABD}} + {\angle {ABD}}}} \\{= {3\angle \; {ABD}}} \\{= {3\; {\theta.}}}\end{matrix}\quad$

Notice that such above digression applies equally as well to varyingmagnitudes of given angle ABD since each successive step listed in suchsequence of Euclidean operations and accompanying proof acts upon priorinformation that commences directly from such angle ABD, no matter whatits initial size. That is to say, according to such proof, angle ABCstill would be of magnitude 3θ, even if given angle ABD were to becomevaried to some degree in overall size. During such conditions, FIG. 1Awould assume reconstituted shapes that depict actual Kempe devicereconfigurations after being articulated to different positions.

In other words, by defining the magnitude of given angle ABDalgebraically, instead of just assigning a singular value to it, suchFIG. 1A thereby describes all of the various attitudes which suchinvention could assume during articulation. Such multiple number ofactual drawings, thereby collectively comprising an entire formulation,essentially consists of an amalgamation of renderings which aregenerated solely by administering a specific sequence of Euclideanoperations to a given angle ABD which is allowed to vary in size while,in each case, ultimately rendering an angle ABC which amounts to exactlythree times the size of such given angle ABD.

As such, the three Kempe anti-parallelograms incorporated into suchmechanism, as represented in FIG. 1A, thereby serve to regulate, orstrictly control the overall movement of such device, whereby trisectingangles thereby become portrayed as angle ABC becomes altered to aninfinite variety of angles that are in need of being trisected.

A second significant trisection development also was reported in such1897 publication entitled The Works of Archimedes wherein it was claimedthat a marked ruler arrangement placed upon a particular geometricdrawing in a prescribed manner could achieve trisection. In order toaccount for the enormous lapse of time which had preceded suchpublication, page cvi therein stipulates that some such similaroperation might have contributed to the development of the conchoid; acurve considered to have been discovered by Nicomedes somewhere between200 BC and 70 BC.

Such unusual application is represented as prior art in FIG. 1B suchthat straight line MR describes the longitudinal centerline of suchmarked ruler device. The mechanism clearly qualifies as a ruler, asopposed to just a straightedge, simply because it contains a notchetched a specific distance away from one of its tips. This is denoted bythe very small circle appearing upon such drawing at point N.

Therein, angle QPS represents an angle that is to become trisected,suitably drawn upon a separate piece of paper and specifically sized tobe of algebraically expressed 30 designed magnitude. Its sides QP and SPboth are drawn to be equal in length to the distance which such notchresides away from the left endpoint of such marked ruler device.Thereafter, a circle is drawn whose center point is located at vertex Pof angle QPS such that its circumference passes both through point Q andpoint S. Lastly, side SP is extended a sufficient length to the left,thereby completing such geometric construction pattern.

In order to trisect angle QPS, such marked ruler device thereby becomesindiscriminately jockeyed above such piece of paper until such time thatall three of the following listed events occur:

its longitudinal centerline overlaps point Q;

its notch aligns upon the circumference of such drawn circle; and

its left endpoint hovers directly over straight line SP extended.

Therein, point N designates the position where such notch hovers abovesuch previously drawn circle.

As validated below, once such device becomes set in this particularmanner, angle RMP constitutes an actual trisector of angle QPS. Itsrespective sides consist of the longitudinal centerline of suchrepositioned marked ruler device, as represented by straight line MR inFIG. 1B, along with straight line SP extended, as previously drawn upona piece of paper that such marked ruler device now rests upon. Therein,trisector RMP is designated to be of magnitude θ, amounting to exactlyone-third the size of previously drawn angle QPS.

As depicted in FIG. 1B, angle RMP can be proven to be an actualtrisector for angle QPS, by considering that both angles constituterespective given and rendered angles belonging to such famous Archimedesproposition.

The proof that FIG. 1B furthermore constitutes an Archimedes propositionis predicated upon the fact that it meets the description afforded inProposition 8 in the Book of Lemmas, properly rephrased to state that ifQN be any chord of a circle whose center is designated as point P, andif QN be produced to M so that NM is equal to the radius; if further MPproduced meets the circle at point S, the arc QS will be equal to threetimes the arc subtended between straight lines PN and PM. Theaccompanying proof thereby is given as:

${ext}.\begin{matrix}{\mspace{11mu} {{{\angle \; {PNQ}} = {2\angle \; {NMP}}},{{{by}\mspace{14mu} {{isos}.\mspace{14mu} \Delta}\; {NMP}};}}} \\{{= {\angle \; {PQN}}},{{{by}\mspace{14mu} {{isos}.\mspace{14mu} \angle}\; {PQN}};}}\end{matrix}$ $\begin{matrix}{\; {{{{{ext}.\mspace{14mu} \angle}\; {QPS}} = {{\angle \; {QMP}} + {\angle \; {PQM}}}},{{{by}\mspace{14mu} \Delta \mspace{14mu} {MPQ}};}}} \\{= {{\angle \; {NMP}} + {\angle \; {PQN}}}} \\{= {{\angle \; {NMP}} + {2\angle \; {NMP}}}} \\{= {3\angle \; {NMP}}} \\{= {3\angle \; {RMP}}} \\{{= {3\angle \; {NPM}}},{{{{since}\mspace{14mu} {NM}} = {NP}};}}\end{matrix}$

and

hence, the arc QS must be equal to three times the arc subtended betweenstraight lines PN and PM.

Since the specific sequence of Euclidean operations which governs suchfamous Archimedes formulation furthermore distinguishes the very samegeometry as now is represented in FIG. 1B, angle RMP and angle QPS, asposed therein, also must describe its respective given and renderedangles.

With particular respect to FIG. 1B, this is demonstrated as follows,wherein each of the seven steps comprising such specific sequence ofEuclidean operations is stipulated directly below:

step 1—given acute ∠RMP, of arbitrarily selected angular size,designated therein as being of magnitude θ, and exhibiting sides ofarbitrarily selected lengths, so long as RM is sufficiently longer thanMP is first geometrically constructed on a piece of paper, or upon adrawing board;

step 2—given ∠RMP is duplicated such that its vertex is positioned atpoint P; whereby straight line PM denotes one of its sides, with itsother side geometrically constructed so that it resides upon the sameside of straight line PM as does remaining side RM of given angle RMP,such that the duplicated angle faces given angle RMP, or opens uptowards it;

step 3—point N becomes designated as the intersection point betweenstraight line RM and the additional side drawn by such duplicated angle,thereby completing the geometric construction of isosceles triangle MNP;

step 4—a circle is drawn whose origin is placed at point P which is ofradius equal in length to straight line PN;

step 5—straight line MP is extended to point S lying upon thecircumference of such formed circle;

step 6—the other intersection point which straight line MR makes withthe circumference of such formed circle is designated as point Q; and

step 7—straight line PQ is drawn.

A three step algebraic proof, serving to verify that such geometricallyconstructed angle QPS is exactly three times the magnitude of givenacute angle RMP, for any and all magnitudes which it otherwise mightassume is provided as follows:

step 1—because ∠PNQ is as an exterior angle of isosceles triangle MNP,it must be equal to the sum of such triangle's equally sized includedangles, denoted therein as ∠NMP and ∠NPM, which algebraically can besummed to θ+θ=2θ;

step 2—since angle NQP and angle PNQ reside opposite the equal lengthsides of isosceles triangle NPQ, with such sides furthermorerepresenting equal length radii of such drawn circle, they must be ofequal magnitude, such that ∠NQP=∠PNQ=20; and

step 3—because ∠QPS qualifies as an exterior angle of triangle MPQ, itthereby must be equal to the sum of such triangle's included angles QMPand MQP which, by furthermore being related in the two identities∠QMP=∠NMP and ∠MQP=∠NQP, can be algebraically summed to amount toθ+2θ=3θ.

Above, it has just been proven that the magnitude of rendered angle QPSalways must amount to exactly three times that of given acute angle RMP,even as such given acute angle becomes varied in size

Since the actual value of given acute angle RMP is not specifiedanywhere in the sequence of Euclidean operations that FIG. 1B is derivedfrom; it can have no bearing, or direct influence upon implementing it.Hence, introducing algebraic nomenclature, as is posed in FIG. 1B,cannot conflict in any way with its administration; thereby enablingsuch sequence of Euclidean operations to be applied many times over inorder to generate a wide range of patterns that result as given acuteangle RMP varies in size.

Along with such marked ruler arrangement, other trisection mechanismswhose designs could be arranged into geometric shapes that either areindicative of such famous Archimedes proposition, or particular adjunctsthereof, such as the configuration depicted in FIG. 1B, hereinaftershall constitute CATEGORY II articulating trisection devices.

A third significant evolution took root during the early 1900's when anample supply of hinged linkage assemblies, replete with interconnectingpivot pins, summarily became incorporated into a broad spectrum ofup-and-coming applications. Such design practice, extending all the wayup until present day, remains of paramount importance because it enablesan innumerable variety of meaningful motions to become portrayed.

As a consequence of such progress, modern day trisection devices cameinto existence shortly after inventors figured out how to shape thelongitudinal centerlines of linkages and radial centerlines ofinterconnecting pivot pins which collectively constitute theirmechanisms into configurations that are indicative of geometricconstruction patterns of rendered angles whose magnitudes amount toexactly three times the sizes of their respective given angles.

The very process of trisection became more sophisticated thereafter byslotting such linkages, thereby affording added degrees of freedom. Thisenabled such designs to travel over an ever increasing range of distincttrajectories that otherwise simply couldn't be duplicated by solidlinkage designs of comparable shapes, the latter of which proved also tobe both heavier and more costly.

A distinct example of such improved design is presented in FIG. 1C,wherein a pair of slotted linkages are featured which can be used toactuate such device by means of rotating linkage MR in either aclockwise or counterclockwise direction relative to member MS about ahinge located at axis M.

FIG. 1C, was chosen to represent a typical example of such slottedlinkage design for the principal reason that the very artwork which isexpressed in FIG. 1B precisely pinpoints the particular placements ofthe longitudinal centerlines of linkages and radial centerlines ofinterconnecting pivot pins which collectively comprise it.

Because of such association, notice that both FIGS. 1B and 1C denote thevery same position letters, being a convenience which should permit fortheir easy comparison.

Whereas a virtually unlimited number of other geometric constructionpatterns which also serve to constitute such Archimedes proposition, asrepresented FIG. 1B, exactly describe a series of repositionedplacements of the longitudinal centerlines of linkages and radialcenterlines of interconnecting pivot pins which collectively comprisesuch mechanism, as depicted in FIG. 1C as it becomes actuated, theangular notations algebraically denoted as θ and 30 in FIG. 1B therebyalso are shown to carry over into FIG. 1C.

Except for these angular notations, such FIG. 1C merely is a truncatedrendition of prior art. Because its overall geometry can be related insuch manner directly to that of FIG. 1B, and thereby furthermore can beassociated to such famous Archimedes proposition, it is said to qualifyas a CATEGORY II articulating trisection device.

This truncated phenomena becomes quite evident once realizing that suchbasic Archimedes proposition, as featured in FIG. 1B, furthermore couldbe supplemented, or added to, merely by means of incorporatingsuccessive isosceles triangles onto it, each of whose angles of equalsize thereby always would amount to the sum of the magnitudes of the nonadjacent included angles of the triangle whose perimeter represents theouter envelope of the combined triangles which preceded it, therebyestablishing the following progression:

θ+θ=2θ;

θ+2θ=3θ;

θ+3θ=4θ; and

θ+4θ=5θ; etc.

The manner in which such progression could be introduced into such priorart, as posed in FIG. 1C, is indicated by the addition of the link, asdepicted by phantom lines, which extends from axis Q therein.

Aside from such superfluous phantom link, upon viewing FIG. 1C, itbecomes obvious that it consists of both solid, as well as slottedlinkages, along with interconnecting pivot pins, which can described ingreater detail as follows:

solid linkages NP and PQ, each of the same length, have circular holesof the exactly the same size drilled through them located very close toeach of their ends along their respective longitudinal centerlines. Allfour holes are located such that the distance between the respectiveradial centerlines of the holes drilled through linkage NP is the sameas that afforded between the respective radial centerlines of the holesdrilled through linkage PQ, thereby describing the same bolt holefootprints;

a pivot pin whose radial centerline is located at axis Q is insertedthrough one of the holes drilled through linkage PQ and then through theslot afforded by linkage MR which thereby constrains its movement suchthat axis Q always lies along the longitudinal centerline of linkage MRduring device articulation; thereby furthermore conforming to thedistinct geometric pattern posed in FIG. 1B wherein point Q is shown toreside upon straight line MR;

the longitudinal centerline of extended slotted linkage SP is alignedwith that of linkage MR by means of drilling matching circular shapedholes at axis M, located very close to the respective endpoints of suchlinkages; whereby a second pivot pin then becomes inserted through suchmatching circular shaped holes. Hence, such axis M conforms to thelocation of point M, as represented in FIG. 1B, designated as theintersection point between straight lines MR and SP extended; and

another hole is drilled through a portion of linkage MR which does notcontain a slot, whose radial centerline both is aligned with thelongitudinal centerline of linkage MR, and is offset a fixed distanceaway from axis M that is equal to the length between the radialcenterlines of respective holes which previously were drilled throughthe respective ends of linkage PQ. A third pivot pin is inserted throughone of the holes drilled through linkage NP, and then through the vacanthole drilled through linkage MR. Lastly a fourth pivot pin is insertedthrough the vacant hole drilled through linkage NP, then through thevacant hole drilled through linkage PQ, and thereafter through the slotof linkage MP.

Based upon such design, lengths MN, NP, and PQ all must be equal; axis Nmust reside along the longitudinal centerline of linkage MR and upon thecircumference of a circle described about axis P of radius PN, and axisP must reside along the longitudinal centerline of slotted linkage MS;thereby conforming to the geometry posed in FIG. 1B, wherein it is shownthat corresponding straight lines MN, NP, and PQ again are all of equallength, point N resides along straight line MR and upon thecircumference of a circle drawn about point P of radius PN, and point Pis situated upon straight line MS.

In conclusion, such device, as represented in FIG. 1C, is considered tobe fully capable of portraying angle RMS as a trisector for a wide rangeof angle QPS designated magnitudes since each discrete settingfurthermore could be fully described by a singular geometricconstruction pattern, such as that which is afforded in such FIG. 1B; asthereby generated by means of imposing a distinct sequence of Euclideanoperations that conforms to that which governs such famous Archimedesproposition.

Therefore, once such device becomes set by means of rotating linkage MRwith respect to linkage MS so that angle QPS, as posed in FIG. 1C,becomes of particular size 30, its associated trisector, represented asan angle RMP that becomes interposed about axis M between the respectivelongitudinal centerlines of linkages MR and MS, automatically becomesportrayed, being of size θ.

Once a phantom linkage, as depicted in FIG. 1C, becomes incorporatedinto such prior art, a one-to-four angular amplification thereby becomesrealized with respect to angle RMS as forecasted by the progressionexpressed above.

Due to such design intricacy, such device as depicted in FIG. 1C becomescapable of trisecting angles of various sizes in rapid succession. Suchdistinct advantage clearly cannot be matched by a marked rulerarrangement that otherwise must perform the repetitious act ofreproducing all of the cumbersome motions considered necessary toachieve trisection each and every time an angle of different magnitudebecomes slated for trisection.

Mechanisms which fall under the grouping entitled, CATEGORY I,sub-classification B articulating trisection devices feature newdesigns, and thereby shall be fully described at a later time.

For the particular contingency that other trisection methods yet mightbecome identified in the near future, consisting of different approachesthan those which govern such proposed CATEGORY I and CATEGORY IIgroupings, it is recommended that they too should be classified intosuitable categories. For example, one yet to be related method fortrisecting angles consists of portraying specific contours thatrepresent a composite of trisecting angles, or aggregate of previouslyestablished trisection points for angles whose magnitudes amount toexactly three times their respective size. It would seem only fitting,then, to group together such types of mechanisms into an entirely newCATEGORY III articulating trisection device classification.

All told then, just one final important question still remains largelyunaddressed, being that: If a unique motion related solution is requiredfor each and every angle of different designated magnitude that intendedto be trisected, then what distinct proofs, or perhaps interrelated setof proofs, would need to be specified in order to substantiate that somenewly proposed mechanism could perform trisection accurately throughoutits entire range of device settings? As soon will become evident, inorder to suitably answer such looming question, a novel methodology, aspredicated upon an extension to a limited conventional Euclideanpractice that alone is incapable of solving such famous classicalproblem of the trisection of an angle would need to be established;thereby making what will appear in the following pages revolutionary,rather than merely evolutionary, in nature!

SUMMARY OF THE INVENTION

A newly proposed articulating trisection invention is about to beformally introduced which consists of four distinct embodiments. Beforethis can occur, however, a comprehensive methodology first needs to beestablished that identifies specific requirements that each of theirconstituent designs should conform to.

Whereas such comprehensive methodology, in turn, then would need to relyupon new definitions, these are furnished directly below:

mathematically limited activity, an operation that cannot be performedwhen complying with all of the mathematical requirements that have beenimposed upon it. The classical problem of the trisection of an anglequalifies as a very good example in this regard;

overlapment point, an intersection point that resides within a geometricconstruction pattern which, although being easily located byconventional Euclidean means with respect to its given set of previouslydefined geometric data, nevertheless cannot be distinguished in suchmanner from the lone vantage point of particular rendered information;

reversible geometric construction pattern, a geometric constructionpattern that is entirely devoid of overlapment points. Its overallconfiguration can be completely reconstituted by means of launching ageometric construction activity which commences exclusively from any ofits rendered information that is under current review. Reversibilityproceeds because contiguous intersection points therein remaindistinguishable, even with respect to such rendered information; therebyaffording a pathway of return that leads all the way back to its givenset of previously defined geometric data;

irreversible geometric construction pattern, a geometric constructionpattern that harbors overlapment points. Its overall configurationcannot be completely reconstituted by means of launching a geometricconstruction activity which commences exclusively from certain rendereddata because an availability of intrinsic overlapment points residingdirectly along such pathway remain impervious to detection solely byconventional Euclidean means;

backtrack, to distinguish intersection points or even a given set ofpreviously defined geometric data within a geometric constructionpattern by means of applying only a straightedge and compass exclusivelyto identified rendered information;

family of geometric construction patterns, an infinite number ofgeometric construction patterns whose overall shapes vary imperceptiblyfrom one to the next; whereby each drawing is entirely unique due to aslight adjustment which is made to the magnitude of a given angle,denoted as θ, that appears in the very first step of a specific sequenceof Euclidean operations from which all of such drawings can beexclusively derived;

representative geometric construction pattern, a distinct drawing whichhas been selected out of an entire family of geometric constructionpatterns to suitably characterize one relative positioning of itsconstituent straight lines, circular arcs, and intersection points;

Euclidean formulation, a practical means for representing an entirefamily of geometric construction patterns, all upon just a single pieceof paper. Such depiction features just a singular representativegeometric construction pattern that furthermore has an unmistakabledouble arrow notated somewhere upon it that distinguishes it apart froma singular geometric construction pattern. Such type of applicationcould be demonstrated very easily merely by means of placing a doublearrow notation upon such prior art, as posed in FIG. 1B. For theparticular case when such double arrow notation appears just above pointN therein and assumes the shape of two circular arcs residing justoutside of the circle drawn about point P, such convention would signifythat as point N moves about the circumference of such circle, straightline NM of length equal to its radius PN would be geometricallyconstructed from each of such newly established N points, therebylocating respective M points somewhere along straight line SP extended;whereby corresponding Q points in turn would be geometrically located bymeans of extending each distinct straight line MN that becomesrespectively drawn. Whereas the magnitude of angle RMP already is showntherein to be denoted algebraically by the Greek letter θ, it can assumevarying sizes; unlike in the unrelated case wherein such drawingotherwise might constitute a singular geometric construction pattern,thereby requiring that such given angle RMP instead be accorded only asingular numerical magnitude;

animation, as it applies to the motion picture industry, furthermorealso pertains to an entire family of geometric construction patternswhose distinct shapes have been organized in consecutive order forpurposes of either being filmed, or quite possibly, being flappedthrough when inserted into a book; thereby projecting the overallillusion of motion;

replication, an accurate simulation of some particular motion whichbecomes observed as a mechanical device becomes articulated, mostgenerally transacted by means of animating an entire family ofconsecutively arranged geometric construction patterns;

fundamental architecture, prominent portions of an articulating devicewhich are designed to stand out more than others. Various techniques canbe employed to accomplish this which include, but are not limited to,coloring such portions differently, incorporating a distinct declivitysuch as a groove into them, or perhaps making them more pronounced sothat they protrude out beyond other device areas. Unless otherwisespecified, the pathway of such fundamental architecture shall map outthe longitudinal centerlines of linkages and radial centerlines ofinterconnecting pivot pins which collectively constitute sucharticulating device; thereby furthermore possibly being distinguished bythe representative geometric construction pattern of a Euclideanformulation;

static image, the projection of a solid body, as it appears at someparticular point in time when viewed from a singular vantage point inspace;

emulation mechanism, a device which features a fundamental architecturethat regenerates a unique static image for each of its finite settingsthat furthermore can be described by a constituent drawing belonging toa particular family of geometric construction patterns;

trisecting emulation mechanism, a specially designed emulation mechanismin which one particular portion of the unique static image which itportrays actually trisects another portion which corresponds to suchdevice setting;

geometric forming process, a novel method for geometrically describingmotion! On the one hand, such newly proposed process is nothing morethan an extension of conventional Euclidean practice in that each of itsconstituent drawings actually is a singular geometric constructionpattern in itself. On the other hand, a geometric forming processremains unique in that it furthermore relates such output through adistinct association of Euclidean commands. For the condition oftrisection, a particular sequence of Euclidean operations becomesspecified that directs how to generate an entire family of geometricconstruction patterns whose rendered angles amount to exactly threetimes the size of their respective given angles. The overall complexitythat is characteristic of a geometric forming process becomes readilyapparent when considering all of the inputs which contribute to itscomposition, as briefly enumerated below:

trisection rationale, a detailed accounting of how to resolve suchpreviously addressed trisection mystery;

improved drawing pretext, a method which abbreviates the rather outmodedprocess of otherwise unsuccessfully attempting to generate a virtuallyunlimited number of geometric construction patterns in order to fullydescribe a discrete motion which fundamentally is considered to consistof a continuum of unique shapes that instead becomes portrayed over afinite period of time in a rather effortless manner. For purposes ofspecifically substantiating that any static image which becomesregenerated by means of properly setting a trisecting emulationmechanism automatically portrays a motion related solution for theproblem of the trisection of an angle, a corresponding geometricconstruction pattern whose rendered angle is of a magnitude whichamounts to exactly three times its given angle can be selected from asuitable Euclidean formulation which furthermore fully describes itsoverall shape; thereby demonstrating that the smaller portion of suchdisplayed static image actually trisects the larger portion which iscalibrated to such setting;

mathematics demarcation, a natural order that can be attributed to allof mathematics; one that just now becomes recognizable as the result ofpartitioning conventional Euclidean practice with respect to such newlyproposed geometric forming process! In effect, such categorizationeffort allows for any singular geometric construction pattern, therebyremaining stationary with respect to the very piece of paper it is drawnupon, to be distinguished apart from an entire group of geometricconstruction patterns which, not only can be related to one anotherthrough a common set of Euclidean commands, but in such mannerfurthermore can describe an overall outline which becomes duly portrayedby an imparted motion at some arbitrarily selected instant during itsduration. Such concept actually does have a precedent; one which alreadywas imposed upon the well known field physics years earlier, wherein:

statics applies to bodies which are either at rest or else are found tobe moving at a constant speed, thereby signifying a specific conditionwhich results only when forces acting upon such bodies are found to bein equilibrium; whereas

dynamics instead is concerned with the motions of accelerating bodies,thereby applying to a particular condition that is experienced only whenforces acting upon such bodies are determined not to counterbalance oneanother;

set of rules, an accounting of how such newly proposed geometric formingprocess should be governed. Similar to the manner in which the very lawsof motion must be interpreted differently when considering statics, asopposed to dynamics real world involvements, so too would the ruleswhich normally apply to the conduct of conventional Euclidean practiceneed to be interpreted differently when instead considering theadministration of such newly proposed geometric forming process. Forexample, when considering the varying shapes that the fundamentalarchitecture of some particular articulating mechanism might becomerepositioned to over a finite period of time, design issues might ariseconcerning whether or not some specific interference might impede suchflexure from being fully executed; and

probabilistic proof of mathematic limitation, an analysis that providesreasoning for how a motion related solution for the problem of thetrisection of an angle can overcome a mathematic limitation whichotherwise cannot be mitigated when attempting to solve the classicalproblem of the trisection of an angle;

rational number, a ratio between two integers; thereby consisting of anumerator (N) and denominator (D) which mathematically combine in orderto be algebraically expressed as N/D. For any rational number (R) thatfurthermore is real, its actual ‘magnitude’ can be viewed! Its precisevalue could be obtained by means of geometrically constructing a righttriangle whose two sides respectively measure N and D units in length;whereby another right triangle that is similar to it then could be drawnsuch that its side which corresponds to that which is D units long insuch other right triangle would measure one unit in length. Hence, anestablished proportion N/D=x/1 would represent how the lengths of thecorresponding sides of such similar triangles would relate to oneanother, such that the side of unknown length, x, as corresponding tothat whose length is N in such other right triangle, would amount to N/Dunits in length; thereby being of rational value. Strictly speaking, arational number cannot be observed, merely because it is a dimensionlessfraction. However in its stead, what can be viewed is a length whosemagnitude actually equals such value. For example, it reasonably couldbe stated that a straight line which measures 13/3 units in length is ofan overall magnitude that can be expressed as a rational number. In suchabove given definition, notice that no indication whatsoever is affordedas to what magnitudes N and D might assume. As such, they could consistof as many digits as necessary in order to solve any given algebraicproblem. Obviously, without restriction, the greater amount of digitspermitted for any evaluation, the greater number of rational numberswould be contained within its overall field. For example, upon acceptingnumerator and denominator integers of only one digit in length, anentire field of lowest common denominator rational numbers would consistof 1, 2, 3, 4, 5, 6, 7, 8, 9, ½, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 1/3, 2/3,4/3, 5/3, 7/3, 8/3, ½, ¾, 5/4, 7/4, 9/4, 1/5, 2/5, 3/5, 4/5, 6/5, 7/5,8/5, 9/5, 1/6, 5/6, 7/6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 8/7, 9/7, 1/8,3/8, 5/8, 7/8, 9/8, 1/9, 2/9, 4/9, 5/9, 7/9, and 8/9, thereby comprisinga total group of 58 rational numbers. However, if another rationalnumber field instead were to be established that admits all N and Dintegers consisting of two or fewer digits, it naturally would consistof many more rational numbers. If such process were to be allowed tocontinue indefinitely to a point where another rational number fieldwere to be permitted to grow without bounds, so would its veryselectivity to a point where the actual capability of such rationalnumber base to estimate numbers which do not belong to it would increasedramatically. Obviously, a limitless rational number grouping of thisnature would enable any threshold of accuracy which might become setwithin an algebraic problem to be met. As a typical example toeffectively demonstrate such affinity, the transcendental number π couldbe estimated accurately, solely as a rational number, down to asignificance of two decimal places, by applying the equationπ_(EST)=R₁=N₁/D₁=4T/L such that T=R₂=N₂/D₂=773/1,000 and the valueL=R₃=N₃/D₃=985/1,000. Such rational number estimate thereby would amountto a value of π_(EST)=4T/L=4(773/1,000)/(985/1,000)=4(773)/(985)=3.14.Such computation indicates that the rational number 3092/985=3.14 wouldprovide an accurate estimate of the actual value of π down to asignificance of two decimal places. What should stir a far greaterinterest, however, is that later it shall be disclosed exactly how toidentify a more detailed rational number, as consisting of many moredigits, that shall estimate the value of 7 down to a significance of tendecimal places. For such case,T=R₂=N₂/D₂=(77,346,620,052)/100,000,000,000) and the valueL=R₂=N₂/D₂=(984,807,753)/(1,000,000,000). By solving the very sameequation, a valueπ_(EST)=4T/L=4(77,346,620,052×10⁻¹¹)/(984,807,753×10⁻⁹) would result,simplifying toπ_(EST)=4(0.77346620052)/(0.984807753)=4(7.7346620052)/(9.84807753)=3.1415926536;thereby matching the actual value of π when estimated to a ten decimalplace accuracy. Hence,309386480208/98480775300=(309,386,480,208)/(98,480,775,300)=3.1415926536is a rational number that represents an exact value of π down to tendecimal places. Now, further suppose that such significance was notconsidered to be satisfactory with regard to some particular problemwhich instead dictated that only a rational number that can estimate thevalue of π down to eleven significant figures would suffice. Naturallythe number determined above would not qualify, amounting to a value of3.14159265365 down to such eleven place significance; whereby an actualvalue of π down to eleven decimal places of accuracy amounts to3.14159265359. Hence, the number above could be adjusted to(309,386,480,202)/(98,480,775,300)=3.14159265359. As such, by means ofcontinuing to apply such process, improved estimates of the value of πcould be realized all the time in order to meet any accuracyrequirements which might become imposed by some specific problem whichneeds to be solved;

quadratic irrational number, as first distinguished by Al-Mahani overone-thousand years ago; now stated to be the magnitude of any lengthwhich can be geometrically constructed from a given length of unityother than that which is of a rational value. Because its value cannotbe a fraction, it only can be described by a string of decimal numbersthat never repeats itself nor terminates, but surprisingly extendsindefinitely. When algebraically expressed, a quadratic irrationalnumber must exhibit at least one radical sign. However, it cannotfeature any root which is a multiple of three, such as a cube root oreven possibly an eighty-first root, because such values cannot bedetermined by means of applying successive Quadratic Formulas that arepermitted to operate only upon either rational numbers and/or quadraticequation root values, as might become determined by such method. Allthat needs to be known in order to geometrically construct a square rootis that upon drawing a right triangle whose sides become algebraicallyexpressed as a and b, the altitude extending to its hypotenuse, c, willdivide such base into two segments denoted respectively as s and (c−s).Hence, due to three similar right triangles which thereby becomedescribed in such manner, two residing inside of such larger initiallydrawn right triangle, a trigonometric relationship of the form sinθ=b/c=s/b thereby could be established. In that the proportion b/ctherein identifies sides belonging to such larger right triangle, theproportion s/b would apply to corresponding sides belonging to thesmaller right triangle whose hypotenuse is of length b. By multiplyingeach side of such resulting equation by the factor bc, the equalityb²=cs becomes obtained. Then by taking the square root of each side, itbecomes apparent that b=√{square root over (cs)}. As various rationalvalues become substituted for c and s therein, the length of side b ofsuch larger right triangle thereby would assume different square rootmagnitudes. So, if it were intended to geometrically construct side b sothat it amounts to √{square root over (3)} units in length, a righttriangle could be drawn whose hypotenuse, c, amounts to 3 units inlength such that the altitude which lies perpendicular to it wouldreside a distance away from either of its ends a total of one unit ofmeasurement; thereby setting the value of s to be one unit long.Accordingly, the value of length b would amount to √{square root over(cs)}=√{square root over (3(1))}=√{square root over (3)} units inlength. In such very same manner, a fourth root of 3, as amounting tothe square root of √{square root over (3)} and algebraically expressedas

${3^{1/4} = {\left( 3^{1/2} \right)^{1/2} = {\sqrt{3^{1/2}} = \sqrt{\sqrt{3}}}}},$

thereafter could be geometrically constructed, merely by means ofdrawing another right triangle which this time instead exhibitsdimensions of c=√{square root over (3)} and s=1, such that

$b = {\sqrt{cs} = {\sqrt{\sqrt{3}(1)} = {\sqrt{\sqrt{3}}.}}}$

As another example, suppose there were an interest to geometricallyconstruct a quadratic irrational number whose magnitude amounts to

$\sqrt{m + {n\sqrt{q}}}.$

One way to accomplish such activity would be to obtain the square rootof two straight line segments whose extremities become attached to forma longer straight line; the first of which amounts to a length that is mtimes the size of an arbitrarily selected length of unity, whereby suchremaining segment thereby would measure n√{square root over (q)} timesthe size of such unit length. For example, take the case when m=2, n=¾,and q=3; whereby such first length, m, would measure 2 units long, withsuch second length, n√{square root over (q)}, would amount to(¾)√{square root over (3)} units in length. Yet another way togeometrically construct a length whose value amounts to

$\sqrt{m + {n\sqrt{q}}}$

would entail drawing a right triangle whose hypotenuse would be of alength, c, such that its sides, a and b, respectively would amount tolengths √{square root over (m)} and

$\sqrt{n\sqrt{q}};$

thereby relating to one another by virtue of a Pythagorean Theorem,which unconditionally would state,

$c = {\sqrt{a^{2} + b^{2}} = {\sqrt{{\sqrt{m}}^{2} + \left( \sqrt{n\sqrt{q}} \right)^{2}} = {\sqrt{m + {n\sqrt{q}}}.}}}$

Then, when m=2, n=¾, and q=3, side a thereby would amount to a value of√{square root over (m)}=√{square root over (2)} with side b being equalto

$\sqrt{n\sqrt{q}} = {\sqrt{\left( {3/4} \right)\sqrt{3}}.}$

A length √{square root over (m)}=√{square root over (2)} very easilycould be geometrically constructed, merely by means of drawing a righttriangle whose sides each would amount to one unit in length; therebymaking its hypotenuse equal to √{square root over (1²+1²)}=√{square rootover (2)}. Moreover, a length of

$\sqrt{\left( {3/4} \right)\sqrt{3}}.$

could be geometrically constructed by means of drawing yet another righttriangle whose hypotenuse would be √{square root over (3)} units longsuch that the altitude which lies perpendicular to it would reside adistance away from either of its endpoints a total of ¾ units ofmeasurement, such that s=¾; thereby determining a length equal to

$\sqrt{cs} = {\sqrt{sc} = \sqrt{\left( {3/4} \right)\sqrt{3}}}$

units in length. Thereafter, a final right triangle could begeometrically constructed whose sides are of respective lengthsa=√{square root over (2)} and

$b = \sqrt{\left( {3/4} \right)\sqrt{3}}$

such that its hypotenuse would amount to a length of

$c = {\sqrt{a^{2} + b^{2}} = {\sqrt{2 + {\left( {3/4} \right)\sqrt{3}}}.}}$

Lastly, for any number which resides underneath a radical sign whosevalue is negative, a complex number would result; thereby invalidatingany possibility that such value might qualify as a quadratic irrationalnumber, as based upon the reasoning that an imaginary number whichcannot exist most certainly could not be geometrically constructed; and

cubic irrational number, any number whose value is neither rational norquadratic irrational. Although such appellation is typical of Al-Mahaniterminology, more specifically it is intended to signify that such typesof numbers can exist only within root sets of ‘cubic’, or higher, orderalgebraic equations, as posed in a single variable, whose coefficientsare comprised exclusively of either rational and/or quadratic irrationalvalues. In particular, this means that cubic equations whosecoefficients consist of just rational and/or quadratic irrational valuescan be used to convert such number types into corresponding triads ofcubic irrational values. Likewise, three properly associated cubicirrational numbers could be combined mathematically to distinguish arational or quadratic irrational number. In sharp contrast, the root setof any second order equation, as expressed in a single variable, cannotcontain cubic irrational values when its coefficients consist solely ofeither rational and/or quadratic irrational values. One way to refutesuch claim would be to identify a pair of cubic irrational values,denoted as x₁ and x₂, which could satisfy the governing requirementsimposed upon such well known parabolic relationship, as algebraicallyassuming the form ax²+bx+c=0=(x−x₁) (x−x₂)=x²−(x₁+x₂)x+x₁ x₂, for thespecific case when its coefficient ‘a’ amounts to a value of unity;whereby such coefficient, b, would amount to a value of −(x₁+x₂), andsuch magnitude, c, would be equal to the product x₁x₂. When making suchtype futile attempt, however, the very difficulty which would beencountered is that whenever a pair of cubic irrational number valuesbecome selected so that the resulting magnitude of their product, x₁x₂,amounts to some specified rational or quadratic irrational value, suchas c=1, the negative of their sums, amounting to −(x₁+x₂), cannot yieldyet another value equal to some stipulated rational or quadraticirrational value of b. This is because any two values belonging to acubic irrational triad which thereafter become reduced into a singlecubic irrational value, still could not be mathematically combined withsuch remaining value in order to calculate such hoped for result;essentially disqualifying the possibility that only two irrationalnumber values, alone could sufficiently allow such conversion activityto take place. In other words, a quadratic equation, as posed in asingle variable, cannot possibly exist which exhibits coefficients thatsolely are of rational value if its root set were intended to beexpressed in terms of π. However, such result could be easily estimated.For example, when letting x₁=7, and x₂=1/π, such specific value of bwould amount to −(x₁+x₂)=−(π+1/π). By means of simply applying theresults obtained above, the value of such coefficient, b, could beapproximated, as amounting to the particular value−[(309,386,480,202)/(98,480,775,300)+(98,480,775,300)/(309,386,480,202)];whereby the magnitude of its coefficient, c, would be x₁x₂=(π) (1/π)=1.Above, what at face value might seem to be a rather unsupportable oreven preposterous contention, when considered in combination with otherprevailing claims, as rendered by various esteemed mathematics experts,who jointly concede that any cubic equation, as posed in a singlevariable, whose coefficients all are of rational values can contain whatin their words, are ‘constructible’ roots only if accompanied by arational root therein, leads to the extraordinary conclusion, as aboutto be revealed to the general public for the very first time, that eachdistinct algebraic equation format type associates rational, quadraticirrational, and/or cubic irrational numbers in a unique manner. That isto say, when certain types of numerical representations become relatedto each other within a particular problem, they can be expressed only bycertain forms of algebraically equations! For example, as claimed above,no matter what rational and/or quadratic irrational values thecoefficients of a parabolic equation, as posed in a single variable,might assume, it still cannot contain cubic irrational roots. Moreover,contrary to any myth which falsely might allege that a cubic irrationalnumber cannot be geometrically constructed, it truly can! Unfortunately,a raging controversy over such matter, even today, still continues topersist! To clear this up, all that needs to be said is that it mighthave been overlooked on past occasions that by simply drawing an angleof arbitrarily selected size, there exists a good chance that it willexhibit trigonometric properties whose values are cubic irrational. Suchclaim is based merely upon the fact that most angles exhibit propertiesof this nature, whereby it would become highly likely that any randomlydrawn angle would assume such trigonometric proportions. A more properlyphrased statement, as substituted in lieu of this prefabrication, wouldbe that a cubic irrational number is not repeatable; essentially meaningthat the probability of being able to geometrically construct an angleof intended value approaches zero. Accordingly, a cubic irrationalnumber more properly could be defined as any dimensionless value, exceptthat which represents the very ‘magnitude’ of any length which could begeometrically constructed with respect to a given length of unity;thereby otherwise qualifying as the magnitude of a particular lengthwhich instead could be portrayed as a motion related solution for theproblem of the trisection of an angle directly alongside such unitlength. One of the most intriguing aspects of trisection concerns itsassociation with any cubic equation that relates a trigonometricproperty of one angle to that of another whose magnitude amounts toexactly three times its size. An often ignored, but necessary startingpoint to account for such association would consist of explaining howsome trisector actually applies to all three roots belonging to suchtype of cubic equation! Such perplexing concern can be easily rectifiedsimply by divulging that when attempting to divide an angle whosedesignated magnitude is denoted as 3θ into three equal parts, not onlydoes an angle, denoted as θ₁, whose magnitude amounts to exactlyone-third of its value constitute its trisector, but so would otherangles denoted as θ₂=θ₁+120°, as well as θ₃=θ₁+240°. This is becausewhen multiplying such other angles, each by a factor of three, equationswould result of 3θ₂=3θ₁+360°=3θ, as well as 3θ₃=3θ₁+720°=3θ. Hence, whenattempting to determine a trisector for a particular angle denoted as3θ, it always should be kept in mind that, not one, but three anglesactually meet such description; namely being, θ₁, θ₂=θ₁+120°, andθ₃=θ₁+240°. From such knowledge, like trigonometric properties for allthree of such angles could be determined whose respective values, suchas cos θ₁, cos θ₂, and cos θ₃, collectively would represent roots forany of such cubic equations. Accordingly, a process to obtain solutionsof this nature would consist of first verifying that a particular cubicequation which is to be assessed meets such format description. Forexample, the famous cubic equation sin (3θ)=3 sin θ−4 sin³ θ does so bybeing of cubic form, and furthermore relating the sine of an angle,denoted as θ, to that of another angle, denoted as 3θ, whose magnitudethereby would amount to exactly three times its size. Secondly, adesignated magnitude of 3θ would have to become determined by means oftaking the arc sine of the corresponding value which is represented inthe particular equation that is to be solved. When a cubic equation ofthe specific form 3 sin θ−4 sin³ θ−½=0 becomes specified, such ½ valuecould be equated to the sin (3θ), thereby determining that 3θ wouldamount to a value which is equal to the arc sine of ½, being 3θ inmagnitude. Thirdly, a value of θ would be calculated, simply by dividingsuch determined 3θ value by three, yielding θ=3θ/3=30°/3=θ₁=10°. Next,the equations cited above would become applied, wherebyθ₂=θ₁+120=10°+120=130°, and θ₃=θ₁+240=10°+240=250°. The roots of suchgiven equation would be sin θ₁=sin 10°, sin θ₂=sin 130°, and sin θ₃=sin250°, any of which could be substituted back into the cubic equation 3sin θ−4 sin³ θ=½ in order to produce the desired result of. Furthermore,the equations θ₂=θ₁+120° and θ₃=θ₁+240°, although being analogous tothose which previously were determined by De Moivre in connection withhis treatment of complex numbers, now have become more restricted in asense that they furthermore are governed by additional common senserules, such as ‘only three roots can a cubic equation contain’. Basedupon such understanding, it becomes thoroughly evident that specificalgebraic formulas, as listed below, that can be applied in order tosuitably convert triads of cubic irrational values into either rationalor quadratic irrational values, and vice versa. Below, such formulas arearranged so that they can be solved when their root set:

products equate to preselected rational and/or quadratic irrationalnumbers which can be substituted into left-hand portions of suchequations:

cos(3θ₁)/4=cos θ₁ cos θ₂ cos θ₃;

sin(3θ₁)/4=sin θ₁ sin θ₂ sin θ₃; and

tan(3θ₁)=tan θ₁ tan θ₂ tan θ₃;

sums equate to preselected rational and/or quadratic irrational numberswhich can be substituted into applicable left-hand portions of suchequations:

0=cos θ₁+cos θ₂+cos θ₃;

0=sin θ₁+sin θ₂+sin θ₃; and

3 tan(3θ₁)=tan θ₁+tan θ₂+tan θ₃; and

sums of paired products equate to preselected rational and/or quadraticirrational numbers which can be substituted into applicable left-handportions of such equations:

−¾=cos θ₁ cos θ₂+cos θ₁ cos θ₃+cos θ₂ cos θ₃;

−¾=sin θ₁ sin θ₂+sin θ₁ sin θ₃+sin θ₂ sin θ₃; and

−3=tan θ₁ tan θ₂+tan θ₁ tan θ₃+tan θ₂ tan θ₃.

A comprehensive methodology, as presented in FIG. 2, evidences acritical role which new discovery plays in the development of trisectingemulation mechanisms. The main purpose of such flowchart is to providean overall accounting of tasks that are required in order to suitablysubstantiate that a newly proposed invention, such as that which isabout to be formally introduced herein, can perform trisectionaccurately over a wide range of device settings.

The reason why FIG. 2 appears, even before four embodiments formallybecome specified, simply is because their detail designs are predicatedupon such input.

As indicated therein, a trisecting emulation mechanism which is deemedto merit the capability to achieve trisection over a wide range ofdevice settings must suitably demonstrate that its proposed designcomplies with all of the provisions specified in a prepared requirementschart.

Moreover, FIG. 2 specifically itemizes pertinent ramifications which areconsidered to underlie the very nature of such plaguing trisectionmystery.

To expound, by beginning at the oval shaped START symbol therein, noticethat four process boxes of rectangular shape are specified in theiterative portion of such diagram just before the upper diamond shapeddecision box location.

More specifically, they furthermore appear as entries in the very firstvery first column of a FIG. 3 Trisection Mystery Iteration ProcessesTable under the heading entitled, PROCESS.

The second column of such FIG. 3 chart provides correct responses underthe heading entitled, CORRECT RESPONSE appearing therein, for each ofsuch four listed processes, as extracted from later discussions.

Within this iteration portion of FIG. 2, the NO arrow departing suchupper diamond shaped decision box signifies that a review process isneeded in order to assure that for whatever identified mathematiclimitation becomes proposed at a particular point in time, sufficientmeans are specified to overcome it.

Moreover, the oval shaped START symbol presented in FIG. 2 also leads tofive inputs that additionally need to be supplied, as indicated byhaving their names listed within parallelograms.

Based upon a detailed understanding of such five inputs, an overallgeometric forming process is to be established from which explicitdetails can be gleaned in order to furnish correct responses for fiverectangular shaped processes itemized to the very right of such FIG. 2flowchart, all leading to a second decision box cited therein.

The reason for preparing a requirements chart is to identify specificinformation that, although mostly lacking from prior art which wouldqualify as CATEGORY I and CATEGORY II articulating trisection devices,is needed nonetheless to suitably substantiate that the design of anynewly proposed articulating invention can perform trisection accuratelyover its wide range of device settings.

Another process box located just to the left of such second decision boxin FIG. 2 discloses that an initially proposed invention might have toundergo a series of refinements in order to satisfy all of theprovisions imposed by such requirements chart.

Whenever it can be substantiated that a newly proposed invention trulymeets all of such stipulated provisions, as imposed in such citedrequirements chart, it is said to thereby qualify as a trisectingemulation mechanism; thereby becoming capable of performing trisectionmerely by means of becoming properly set to some designated size!

In so doing, its constituent fundamental architecture thereby becomesreconfigured; causing the regeneration of a static image that portraysits trisector!

Trisection occurs because such regenerated static image must bedescribable by a drawing that is part of a distinct family of geometricconstruction patterns, each rendered angle of which is of a magnitudewhich amounts to exactly three times the size of its given angle;whereby the portion of such regenerated static image which correspondsto the given angle in such drawing actually portrays a trisector for itsrendered angle portion, corresponding to a specific setting which suchdevice initially becomes set to.

In FIG. 4, the four embodiments which collectively constitute such newlyproposed invention are individually tabulated, each appearing as aseparate line item under the heading entitled, NEWLY PROPOSEDARTICULATING TRISECTION INVENTION EMBODIMENT NAME.

The second column therein, accorded the heading entitled, APPLICABLEFIGURE NUMBER OF CORRESPONDING EUCLIDEAN FORMULATION, identifies acorresponding Euclidean formulation for each of such listed fourembodiments, as cited in the first column therein.

A good starting point when referring to such Euclidean itemizedformulations is to identify which angles are given and which arerendered therein. In order to expedite such activity, listings of suchangles are presented in a FIG. 9 Euclidean Formulation Rendered AngleRelation Table. First column entries cited therein under the headingentitled, APPLICABLE FIGURE NUMBER OF EUCLIDEAN FORMULATION, reiteratethose listings appearing in the second column of FIG. 4.

Notice therein that each GIVEN ANGLE(S) entry, as it appears in thesecond column of such FIG. 9 table, describes a particular given angle,or pair of given angles when both acute as well as obtuse angletrisection fall under consideration. Moreover, the magnitude of eachcited given angle is algebraically expressed on a line appearingdirectly below it.

Clearly, the very same format applies for each entry listed in the thirdcolumn therein under the heading entitled, RENDERED ANGLE(S).

As an example of this, for the Euclidean formulation presented in FIG.5, as given angle VOO′ of algebraically expressed magnitude θ variesfrom zero to thirty degrees in size, acute angle VOU′ of magnitude 3θ,always amounts to exactly three times its size, thereby varying fromzero to ninety degrees.

In this regard, FIG. 5 represents the very first attempt to describe anentire Euclidean formulation. Its double arrow notation signifies thatsuch improved drawing pretext characterizes an entire family ofunrevealed geometric construction patterns, in addition to the lonerepresentative geometric construction pattern that is depicted upon itsvery face.

Each and every one of such distinct drawings could be geometricallyconstructed merely by means of executing all of the commands which arespecified in its governing fourteen step sequence of Euclideanoperations, enumerated as follows:

step 1—given acute angle VOO′ of arbitrarily selected magnitude θranging anywhere from zero to thirty degrees is geometricallyconstructed such that its side OO′ exhibits the same length as side OV;

step 2—side OV of given acute angle VOO′ becomes designated as the+x-axis;

step 3—a +y-axis is generated orthogonally to such +x-axis, representedas a straight line drawn through vertex O of given angle VOO′ which isgeometrically constructed perpendicular to such designated x-axis;

step 4—a portion of the circumference of a circle denoted as the FIRSTCIRCULAR PORTION is geometrically constructed about center point O whoseradius is equal in length to straight line OV, thereby enabling it topass through points V and O′, both of which previously have beendesignated as respective termination points of angle VOO′;

step 5—the intersection between such FIRST CIRCULAR PORTION and suchy-axis becomes designated as point T;

step 6—next, an angle which amounts to exactly three times the magnitudeof given angle VOO′ becomes geometrically constructed, but in acompletely different, yet simplified manner to the way in which it wasdrawn in FIG. 1B; one which is more in line with the fan shape, as wasgenerated in FIG. 1A. This is to be achieved merely by geometricallyduplicating given angle VOO′ twice, and thereafter adding such resultonto it in order to obtain a new angle of magnitude 3θ such that itsvertex is situated at point O, its first side resides along the +x-axis,and its yet undesignated other side is orientated counterclockwise toit;

step 7—the intersection point between such FIRST CIRCULAR PORTION andthe remaining, yet undistinguished, side of such geometricallyconstructed angle of size 3θ becomes designated as Point U′;

step 8—straight line TU′ and straight line TO become drawn, therebycompleting isosceles triangle TOU′;

step 9—a portion of the circumference of a circle denoted as the SECONDCIRCULAR PORTION is drawn about point O′ whose radius is set equal inlength to straight line OO′;

step 10—the yet undistinguished intersection point which such SECONDCIRCULAR PORTION makes with such previously drawn y-axis becomesdesignated as point T′;

step 11—straight line O′T′ becomes drawn;

step 12—a portion of the circumference of a circle denoted as the THIRDCIRCULAR PORTION is drawn about point T′ whose radius is set equal inlength to straight line TU′;

step 13—the yet undistinguished intersection point which such THIRDCIRCULAR PORTION makes with such SECOND CIRCULAR PORTION now becomesdesignated as point U; and

step 14—straight lines T′U and O′U become drawn, thereby completingisosceles triangle T′O′U.

Any configuration which could be generated when implementing suchsequence of Euclidean operations would exhibit a unique shape based uponthe particular magnitude that becomes assigned to given angle VOO′ inits step 1.

The configurations that such FIG. 5 Euclidean formulation would assumewhen given angle VOO′ adopts its limiting values are specified in detailas follows:

when given angle VOO′ is designated to be of zero degree magnitude:

radii OO′, OU′ and O′T′ all collapse onto the +x-axis;

isosceles triangle OTU′ becomes a right triangle whose hypotenuse TU′furthermore can be described as straight line TV;

isosceles triangle ‘OT’U becomes a right triangle whose hypotenuse UT′furthermore can be described as straight line UO; and

when given angle VOO′ is designated to be of thirty degree magnitude:

isosceles triangle O′OT′ becomes an equilateral triangle whose vertex T′coincides with vertex T;

isosceles triangle OTU′ becomes a straight line which resides upon the+y-axis, such that its side, represented as radius OU′ collapses uponits other side, therein represented as radius OT; and

isosceles triangle O′T′U becomes a straight line, such that its side,represented as radius O′U emanating from a center point O′ collapsesupon its other side, therein represented as radius O′T′.

More particularly, this means:

when given angle VOO′ amounts to 0°, such Euclidean formulation assumesthe form of three sides of a square along with its diagonals comprisedof straight lines TU′ and T′U since point T′ collapses onto point O, andpoints U′ and O′ collapse onto point V; and

when given angle VOO′ is 30, such Euclidean formulation assumes the formof an equilateral triangle one of whose sides aligns upon the +y-axiswith points T′, U, and U′ collapsing onto point T.

Algebraically, such determination is verified below:

when given angle VOO′, denoted as θ in FIG. 5, amounts to 0°,

∠TOU′=UO′T′=90−3θ=90°;

∠TOO′=∠UO′O=90−θ=90°;

∠U′OO′=∠T′O′O=2θ=0°;and

when given angle VOO; amounts to 30°,

∠TOU′=∠UO′T′=90−3θ=0°;

∠TOO′=∠UO′O=90−θ=60°;

∠U′OO′=∠T′O′O=2θ=60°.

Accordingly, the insertion of such double arrow notation into FIG. 5signifies that as the magnitude θ of given angle VOO′ becomesinfinitesimally increased from zero to thirty degrees in the very firststep of such sequence of Euclidean operations, the overall shape of suchfigure will change by means of reconfiguring itself as the result ofpoint T′ becoming displaced upwards from point O to point T along the+y-axis. Once a virtually unlimited number of geometric constructionrenderings which belong to such distinct family of geometricconstruction patterns eventually become drawn, a complete Euclideanformulation finally would be represented.

The particular placement of such vertical double arrow in FIG. 5 is tosignify that point T′ can reside only upon such +y-axis. Such geometricalignment easily can be confirmed once realizing that since radius O′T′is of the same length as radius O′O, as angle VOO′ becomes varied insize, point T′ always must intersect such +y-axis at the juncture of thecircumference of a circle of radius O′O which becomes drawn about pointO′.

Each unrepresented, but differently shaped geometric constructionpattern that also belongs to the family of geometric constructionpatterns which constitute such Euclidean formulation, as represented inFIG. 5, must be structured from the very same sequence of Euclideanoperations whereby:

all unrepresented straight lines corresponding to those which appear assolid straight lines exhibited upon the representative geometricconstruction pattern of such Euclidean formulation, as posed in FIG. 5,must be equal to their respective lengths; and

all unrepresented internal angles that apply to such Euclideanformulation must maintain the same proportions with respect to eachother as appear in its representative geometric construction pattern, asactually is posed in such FIG. 5. For example the magnitudes of angleO′OT′ and angle O′T′O always must remain equal to each other, even whentheir relative sizes become varied, since they represent angles thatreside opposite the equal length sides of isosceles triangle O′T′O.

TU′ and T′U are depicted as phantom straight lines therein to indicatethat their respective overall lengths are permitted to vary from onedrawing to the next within such specific family geometric constructionpatterns.

Such Euclidean formulation, as posed in FIG. 5, is shown to consist ofthree principal portions, identified as follows:

isosceles triangle TOU′, as denoted by darker shading, along with+x-axis;

isosceles triangle T′O′U, as denoted by lighter texture; and

straight line OO′ which interconnects the lower vertices of isoscelestriangle TOU′ and isosceles triangle T′O′U together.

Within FIG. 5, notice that vertex U′ belonging to isosceles triangleTOU′ aligns with, or superimposes directly upon, side O′U of isoscelestriangle T′O′U. This is verified by the following proof:

since the whole is equal to the sum of its parts,

∠VOO′+∠O′OU′=∠VOU′

θ+∠O′OU′=3θ

∠O′OU′=2θ;

∠U′OO′=2θ;

OT=OU′=OO′ because point T, point U′ and point O′ all reside upon suchFIRST CIRCULAR PORTION;

O′O=O′T′=O′U because point O, point T′ and point U all reside upon suchSECOND CIRCULAR PORTION;

since OO′ is equal in length to O′O by identity, whereby viasubstitution OT=O′T′ and OU′=O′U;

T′U=TU′ by geometric construction;

isosceles triangle TOU′ must be congruent to isosceles triangle T′O′Usince their three corresponding sides are of equal lengths;

since the magnitudes of angles included in isosceles triangle T′O′U mustbe of equal respective sizes to corresponding angles featured in itscongruent triangle TOU′, then it can be said that ∠T′O′U=∠TOU′;

since the whole is equal to the sum of its parts,

∠VOU′+∠U′OT=90

∠U′OT=90−∠VOU′

∠TOU′=90−∠VOU′

∠TOU′=90−3θ;

since the whole is equal to the sum of its parts, and by substitution ofthe identities

∠O′OV=∠VOO′=θ and ∠O′OT=∠TOO′,

∠VOO′+∠O′OT=90

∠O′OV+∠TOO′=90

∠TOO′=90−∠O′OV

∠T′OO′=90−θ;

whereby the angles residing opposite the equal sides of isoscelestriangle OO′T′ must be of equal magnitude, such that ∠O′T′O=∠T′OO′;

by substitution, the value of angle O′T′O is equal to 90−θ;

since the sum of the included angles in isosceles triangle OO′T′ must be180 degrees, via substitution,

∠O′T′O+∠T′OO′+∠OO′T′=180

(90−θ)+(90−θ)+∠OO′T′=180

∠OO′T′=180−2(90−θ)

∠OO′T′=2θ;

since the whole is equal to the sum of its parts and via substitutionfrom above,

∠OO′T′+∠T′O′U=∠OO′U

2θ+∠TOU′=∠OO′U

2θ+(90−3θ)=∠OO′U

90−θ=∠OO′U;

whereas the angles residing opposite the equal sides of isoscelestriangle U′OO′ must be of equal magnitude, then ∠OO′U′=∠O′U′O;

since the sum of the internal angles of isosceles triangle OO′U′ mustequal 180 degrees, via substitution,

∠OO′U′+∠O′U′O+∠U′OO′=180

OO′U′+∠OO′U′+∠U′OO′=180

2(∠OO′U′)+2θ=180

2(∠OO′U′)=180−2θ

∠OO′U′=90−θ;

since both angle OO′U′, as well as angle OO′U are equal to a magnitudeof 90−θ, they must be equal in size to each other; and

hence, point U′ must reside on straight line O′U.

For the actual representative geometric construction pattern, as isexpressed upon the very face of the Euclidean formulation representedFIG. 5, given acute angle VOO′ amounts to exactly 16°. As such, angleVOU′ must be exactly three times that size, or 48°, and angle TOU′ mustbe equal to its complement, being 42°.

Accordingly, such algebraic proof further validates that even if such16° given acute angle VOO′, as really is depicted in FIG. 5, were tohave been of slightly different size, point U′ nevertheless would residesomewhere along straight line O′U. This is because the very samesequence of Euclidean operations would have governed the development ofanother algebraic proof for such altered case, whereby a distinctdrawing of somewhat modified overall shape, but one which nonethelessbelongs to the very same family of geometric construction patterns,instead would have replaced the representative geometric constructionpattern that presently is depicted upon the very face of such Euclideanformulation.

Moreover, since such argument furthermore applies to any distinctoverall shape otherwise contained within such distinct family ofgeometric construction patterns, such alignment of point U′ alongstraight line O′U thereby must pertain to any and all of such drawingswhich collectively comprise it.

Since such representative geometric construction pattern, as actually isdepicted in FIG. 5, very easily could be appended simply by means ofincorporating additional steps onto its sequence of Euclideanoperations, it furthermore becomes possible to devise a never-endingassortment of Euclidean formulations which stem directly from it.

Accordingly, a derivative Euclidean formulation, as represented in FIG.6, whose sequence of Euclidean operations builds upon that which wasapplied to develop such Euclidean formulation, as represented in FIG. 5,by means of appending another three steps onto it, thereby is said toconsist of a lengthened seventeen step sequence of Euclidean operationswhose last three steps are provided as follows:

step 15—straight line U′O is extended downwards and to the left to aposition where its meets such FIRST CIRCULAR PORTION, whereby suchintersection point becomes designated as point W;

step 16—straight line OO′ is perpendicularly bisected such that theposition where its downward extension intersects such FIRST CIRCULARPORTION becomes designated as point X; and

step 17—straight line OX becomes drawn.

FIG. 7 depicts a second derivative Euclidean formulation, therebyserving as an example of how to create others. Therein, member notationsand shadings that are specified upon prior Euclidean formulations havebeen omitted because they no longer are needed. However, in theirabsence, a rectangle appears whose upper two corners align with point T′and point U′ therein. Such drawing can be geometrically constructed bymeans of appending the seventeen step sequence of Euclidean operationswhich such derivative Euclidean formulation, as posed in FIG. 6, wasgenerated from into a somewhat larger twenty-one step sequence ofEuclidean operations by incorporating steps 18 through 21 onto it asfollows:

step 18—straight line T′U′ is drawn;

step 19—a straight line passing through point T′ is drawn perpendicularto straight line OO′;

step 20—an additional straight line passing through point U′ is drawnperpendicular to straight line OO′; and

step 21—the intersection between radii OU′ and O′T′ is designated aspoint Y.

The proof that such newly drawn straight line T′U′ runs parallel toradius OO′ for all magnitudes which given angle VOO′ could assume, isprovided directly below:

since angle OO′T′ and angle U′OO′, as included in triangle OO′Y, bothare equal to a magnitude of 2θ, such triangle must be isosceles, wherebytheir opposite sides OY and O′Y must be of equal length;

whereas, straight line OO′ constitutes a radius belonging to both suchFIRST CIRCULAR PORTION and SECOND CIRCULAR PORTION, as posed in FIG. 7,radii OU′ and O′T′, by being equal in length to it, must be equal inlength to each other;

since the whole is equal to the sum of its parts, via substitution fromabove,

OU′=O′T′

OY+YU′=O′Y+YT′

OY+YU′=OY+YT′

YU′=YT′;

hence, triangle T′YU′ must be isosceles:

whereas angle OYO′ and its vertical angle T′YU′ must be of equalmagnitude, and the sum of the internal angles of a triangle must equal180°, it can be stated for isosceles triangles OYO′ and T′YU′ that

$\begin{matrix}{{180{^\circ}} = {{\angle \; T^{\prime}{YU}^{\prime}} + {2\left( {\angle \; {YU}^{\prime}T^{\prime}} \right)}}} \\{= {{\angle \; {OYO}^{\prime}} + {2\left( {\angle \; {YU}^{\prime}T^{\prime}} \right)}}} \\{{= {{\angle \; {OYO}^{\prime}} + {2\left( {\angle \; {YOO}^{\prime}} \right)}}};}\end{matrix}\quad$

hence, since ∠YU′T′ must be equal in magnitude to ∠YOO′, straight lineT′U′ therefore must be parallel to radius OO′ because radius OU′, bymeans of acting as a transversal, distinguishes such angles to bealternate interior angles of equal magnitude with respect to each other.

The double arrow convention, as depicted in such second derivativeEuclidean formulation posed in FIG. 7, indicates that point T′intersects the +y-axis upon all drawings which belong to itsdistinguished family geometric construction patterns. This is becausethe addition of such rectangle has no bearing whatsoever upon theoutcome posed by such derivative Euclidean formulation, as depicted inFIG. 6, because it is represented only by phantom lines which either cangrow or shrink in size as the magnitude of given angle VOO′ becomesaltered in variable size, denoted as θ.

A third derivative Euclidean formulation, as posed in FIG. 8, buildsupon such previously described twenty-one step sequence of operations.However, because angle VOU′, while still amounting to a magnitude of 3θtherein, nevertheless is to be geometrically constructed in an entirelydifferent manner, the following steps do not apply to FIG. 8:

steps 6-8; and

steps 12-21;

Instead, the following additional steps complete the sequence ofoperations for such third derivative Euclidean formulation:

step 22—straight line OO′ is perpendicularly bisected such that theposition where its upward extension intersects straight line O′T′becomes designated as point Y;

step 23—a radius is drawn which emanates from center point O, passesthrough point Y, and terminates at a location upon such FIRST CIRCULARPORTION which becomes designated as point U′;

step 24—straight line O′T′ is extended to a position where itsintersects such FIRST CIRCULAR PORTION, thereafter designated as pointZ;

step 25—straight line OZ is drawn;

step 26—phantom line U′Z becomes drawn; and

step 27—orthogonal transformed axes x_(T) and YT are geometricallyconstructed with point O furthermore designating their origin, such thatthe +y_(T) axis superimposes upon radius OU′, as previously drawntherein.

An accounting of such additional steps is furnished as follows:

step 22 above replaces steps 16 and 21; and

step 23 above replaces step 7.

To summarize, the entire sequence of operations from which such thirdderivative Euclidean formulation, as posed in FIG. 8, was developedconsists of the following steps:

steps 1-5;

steps 9-11; and

steps 22-27.

A reconciliation of the various angles appearing in FIG. 8 is providedbelow:

whereas it previously was proven that angle OO′T′ is of magnitude 2θwhenever the magnitude of given angle VOO′ is algebraically designatedto be θ, once furthermore realizing that triangle OO′Y must be isoscelesbecause its vertex Y resides upon the perpendicular bisector of its baseOO′, it must be that angle VOU′ amounts to a magnitude of 3θ because,

$\begin{matrix}{{\angle VOU}^{\prime} = {{\angle \; {VOO}^{\prime}} + {\angle \; O^{\prime}{OU}^{\prime}}}} \\{= {{\angle \; {VOO}^{\prime}} + {\angle \; O^{\prime}{OY}}}} \\{= {{\angle \; {VOO}^{\prime}} + {\angle \; {OO}^{\prime}Y}}} \\{= {{\angle \; {VOO}^{\prime}} + {\angle \; {OO}^{\prime}T^{\prime}}}} \\{= {\theta + {2\; \theta}}} \\{{= {3\; \theta}};}\end{matrix}\quad$

moreover, with straight line OZ being geometrically constructed to be ofequal length to straight line OO′, triangle OO′Z also must be isosceles.Therefore its included angle ZOO′ must be of magnitude 180−4θ by way ofthe fact that ∠OO′T′=φOO′Z=∠O′ZO=2θ. Also, since angle VOO′ is equal toθ, with the whole being equal to the sum of its parts:

∠ VOZ = ∠ VOO^(′) + ∠ O^(′)OZ $\begin{matrix}{{\angle \; {VOZ}} = {{\angle \; {VOO}^{\prime}} + {\angle \; {ZOO}^{\prime}}}} \\{= {\theta + \left( {180 - {4\theta}} \right)}} \\{= {180 - {3\; {\theta.}}}}\end{matrix}$

Thus, the angle which is supplementary to obtuse angle VOZ must be equalto 3θ. Once the length of straight line OZ becomes algebraicallyexpressed by the Greek letter ρ, and the sin (3θ) becomes designated bythe Greek letter η, point Z thereby must reside a vertical distance ofpf above the x-axis. This same condition naturally applies for point U′since angle VOU′ also is of magnitude 3θ with straight line OU′ alsohaving been geometrically constructed to be of equal length to straightline OO′.

Hence, phantom line U′Z must remain parallel to the x-axis, even duringconditions when such given angle VOO′ of designated magnitude θ variesin size, as it is capable of doing in such third derivative Euclideanformulation, as is represented in FIG. 8. This is because the verticaldistances both being dropped from point Z and point U′ to an x-axis,furthermore represent projections of length pf. Such projections veryeasily furthermore could be considered as opposite equal length sides ofa rectangle, thereby imposing the requirement that phantom line U′Z mustremain parallel to such x-axis at all times.

As such, once the double arrow convention becomes applied to FIG. 8,such illustration may be construed to be the Euclidean formulation of anentire family of geometric construction patterns in which angle VOU′must be of size 3θ and angle VOZ must be of magnitude 180−3θ whenevergiven angle VOO′ is of designated magnitude θ; such that point U′ andpoint Z must reside the same distance above the x-axis as for allgeometric construction patterns that belong to such family.

As given angle VOO′ varies in size, because each intersection point Ybetween straight line OU′ and straight line O′T′ furthermore must resideupon the perpendicular bisector of straight line OO′, such completefamily of geometric construction patterns, as depicted in FIG. 8, can befilmed in consecutive order for purposes of replicating the motion of acar jack as it otherwise either could be raised or lowered whenattempting to change a tire.

Hence, it could be said that a car jack configuration of such designcould regulate angle VOU′ so that its size always amounts to exactlythree times the size of given angle VOO′ during conditions when itsmagnitude becomes varied.

The actual mechanics behind such activity simply is that:

as given angle VOO′ of designated magnitude θ becomes adjusted in size,angle OO′T′ always must be equal to double its size, amounting to amagnitude of 2θ due to the geometric construction of isosceles triangleOO′T′ thereby exhibiting two angles of (90−θ) size; such that

once straight line OU′ is located so that it emanates from center pointO and passes through point Y, being the intersection point betweenstraight line O′T′ and the perpendicular bisector of straight line OO′,the latter of which furthermore represents the base of isoscelestriangle OO′Y, then angle O′OU′ which thereby becomes described alsomust be equal in size to angle OO′T′, amounting to a magnitude 2θ; suchthat

angle VOU′ must be equal to the sum of the magnitudes of such givenangle VOO′ plus angle O′OU′ which calculates to θ+2θ=3θ.

Moreover, notice that such geometry additionally featuresanti-parallelogram OU′O′T′, as described by the fact that its diagonalsOU′ and O′T′ are of the same length and intersect at point Y, which islocated upon a perpendicular bisector of a straight line OO′ thatconnects two of the endpoints of such diagonals together. Proof of thislies in the understanding that such description can occur only whentriangle YOT′ is congruent to triangle YO′U′, such that the otherrequirement of being an anti-parallelogram, being that its oppositesides OT′ and O′U′ are of equal length also becomes fulfilled. Suchproof relies upon a side-angle-side (SAS) determination wherein:

straight line YO and straight line YO′ of isosceles triangle YO′ must beof equal length;

angle T′YO must be equal in magnitude to vertical angle U′YO′; and

straight line YT′ and straight line YU′ must be of equal length becausethey complete respective straight line O′T; and straight line OU′, alsobeing of equal lengths.

The actual motion of such anti-parallelogram OU′O′T′, as furthermorereplicated by means of animating, in consecutive order, the uniquedrawings which belong to the distinct family of geometric constructionpatterns which is represented by such third derivative Euclideanformulation, as posed in FIG. 8, is entirely different from that whichotherwise would be portrayed by such articulating such Kempeanti-parallelogram construction, as it appears in the prior art formerlydepicted in FIG. 1A; principally because the two opposite sides of suchanti-parallelogram OU′O′T′, appearing previously as straight linesegments OT′ and O′U′ in FIG. 7, although always remaining of equallength to each other, nonetheless must vary in size during flexure.

This becomes clear by further examining FIG. 8 and noticing that asgiven angle VOO′ varies in magnitude, point T′ must move verticallyalong the y-axis whereby, as straight line segment OT′ changes inoverall length, the straight line distance between point O′ and point U′also must adjust accordingly to be equal to such length.

Accordingly, such car jack arrangement, while preserving the features ofa previously described anti-parallelogram, by means of removing is sidemembers enables the distances that become interposed therein to varywhile remaining of equal length.

FIGS. 6, 7 and 8 thereby represent just three examples of how thedistinct sequence of Euclidean operations which such unique Euclideanformulation was predicated upon, as posed in FIG. 5, furthermore couldbe appended and/or modified in order to establish additional Euclideanformulations of entirely different overall compositions.

FIGS. 5, 6, 7 and 8 essentially form a network of Euclidean formulationswhich allows for a wider base of unique embodiments to thereby becomeprescribed, all of which become capable of trisecting angles in entirelydifferent ways.

Notice that such double arrow convention is notated in all of suchEuclidean formulations. In FIG. 8, such notation is indicative of thefact that as given angle VOO′ varies in size, phantom straight line U′Z,as well as phantom straight line T′Z (being an extension of straightline O′T′) thereby represent adjustable lengths.

In order to provide a motion related solution for the problem of thetrisection of an angle, a particular mechanism could be devised to haveits fundamental architecture regenerate a static image thatautomatically portrays a trisector for a singular angle of anydesignated magnitude which such device could be set to.

In the event that such singular angle turns out to be acute, itsmagnitude algebraically could be denoted as 3θ by considering θ≤30°.Then, for a condition in which such singular angle instead turns out tobe obtuse, its size would be expressed as a supplemental value, therebybecoming algebraically denoted as either 180-3θ, or 270-6θ.

By means of choosing a suitable Euclidean formulation from such FIG. 9Euclidean Formulation Rendered Angle Relation Table that refers directlyto such determinable algebraic expression, a drawing could be identifiedout of its vast family of geometric construction patterns whose renderedangle value matches the very magnitude of such singular designatedangle.

Then, in the event that the static image which becomes regeneratedfurthermore could be fully described by such identified drawing, itsportion which corresponds to the given angle of such drawingautomatically would portray a bona fide trisector for such devicesetting.

Serving as an example of such rather cumbersome logic, a particulardevice of such type is to be set to a designated magnitude of 123.3°.

Since 90°≤123.3°≤180°, such designated angle would qualify as beingobtuse whereby:

in one case 180°−3θ=123.3°

60°−θ=41.1°

−θ=41.1°−60°

θ=18.9°;

for the static image that becomes regenerated in such case toautomatically portray a trisector of 41.1° magnitude, according to suchFIG. 9 Euclidean Formulation Rendered Angle Relation Table, the overallshape of such angle, thereby algebraically expressed therein as being of60°−θ size, would have to match that which appears within a uniquegeometric construction pattern that could be drawn by commencing from agiven angle of VOO′ of 18.9° magnitude, upon which becomes executed allof the remaining commands which are specified in the distinct sequenceof Euclidean operations for either of such Euclidean formulations, asposed in FIG. 6 and FIG. 7. Secondly, for such other case 270°−6θ=123.3°

90°−2θ=41.1°

−2θ=41.1°-90°

2θ=48.9°

θ=24.45θ; and

for the static image that becomes regenerated in such other case toautomatically portray a trisector of 41.1° magnitude, according to suchFIG. 9 Euclidean Formulation Rendered Angle Relation Table, the overallshape of such angle, thereby algebraically expressed therein as being of90°−2θ size, would have to match that which appears within a uniquegeometric construction pattern that could be drawn by commencing from agiven angle of VOO′ of 24.45° magnitude upon which becomes executed allof the remaining commands which are specified in the distinct sequenceof Euclidean operations for such third derivative Euclidean formulation,as posed in FIG. 8.

Although featuring unique control mechanisms, CATEGORY Isub-classification B articulating trisection devices nevertheless can begrouped together because they all feature similar fan array designs.This can be verified merely by means comparing their individual designsto one another. The results of such activity are presented in FIG. 10.

Therein, headings entitled, . . . EMBODIMENT NAME, AND . . . FIGURENUMBER OF CORRESPONDING EUCLIDEAN FORMULATION summarize listings thatappear in FIG. 4.

Radii listings, appearing in groups of three, as cited in the secondcolumn of such FIG. 10 Category I Sub-classification B Conforming AspectChart, under the heading entitled, FAN PORTION RADIUS LISTINGS, alignupon the longitudinal centerlines of linkages that comprise the spokesof such fan arrays. The third column therein, as headed by the words FANPORTION COMMON INTERSECTION POINT LISTINGS, is devoted to identifyingcommon intersection points which align upon the radial centerlines ofinterconnecting pivot pins that are located at the respective hubs ofsuch fan arrays.

For CATEGORY I, expanded sub-classifications definitions are providedbelow, as are premised upon new terminology which previously wasfurnished at the outset of this section:

a CATEGORY I, sub-classification A device hereinafter shall regarded tobe any articulating trisection device which features four linkages ofequal length, excepting that lengths of double that size also arepermissible, all hinged together about their longitudinal centerlines byan interconnecting pivot pin that is passed through one end portion ofeach such that its radial centerline aligns upon the common meetingpoint of such linkage longitudinal centerlines, or instead is passedthrough the center portion of a linkage which is twice such length;thereby collectively constituting the array of a fan which, incombination with the longitudinal centerlines of linkages and radialcenterlines of interconnecting pivot pins which collectively compriseits incorporated unique control mechanism, features a fundamentalarchitecture that is capable of regenerating a multitude of staticimages over a wide range of device settings; whereby such automaticallyportrayed overall outlines furthermore can be described by an entireEuclidean formulation which can distinguish a central angle that amountsto the size of any designated angle which can be set into such device,as subtended between two radii of a circle which thereby becomes dividedinto three equal angular portions by two other radii; and

a CATEGORY I, sub-classification B device hereinafter shall be regardedto be any articulating trisection device which features three linkagesof equal length, excepting that lengths of double that size also arepermissible, all hinged together about their longitudinal centerlines byan interconnecting pivot pin that is passed through one end portion ofeach such that its radial centerline aligns upon the common meetingpoint of such linkage longitudinal centerlines, or instead is passedthrough the center portion of a linkage which is twice such length,thereby collectively constituting the array of a fan which, incombination with the longitudinal centerlines of linkages and radialcenterlines of interconnecting pivot pins which collectively compriseits incorporated unique control mechanism, features a fundamentalarchitecture that is capable of regenerating a multitude of staticimages over a wide range of device settings; whereby such automaticallyportrayed overall outlines furthermore can be described by an entireEuclidean formulation which can distinguish a central angle that amountsto the size of any designated angle which can be set into such device,as subtended between two radii of a circle which thereby becomestrisected by another radius.

All in all, a total of five individual requirements, as belonging to thechart posed in the lower right hand portion of FIG. 2, need to besatisfied before the design of a proposed articulating inventionactually can qualify as a trisecting emulation mechanism. Listed below,these consist of:

RQMT 1—identifying which particular settings, or range(s) thereof, suchdevice is supposed to trisect. Providing such details should disclosewhether acute, as well as obtuse angles apply;

RQMT 2—stating the reason the classical problem of the trisection of anangle cannot be solved. Providing such details should unmask a Euclideanlimitation that needs to be mitigated;

RQMT 3—indicating how such device is to be operated. Providing suchdetails should disclose whether such proposed articulating inventionneeds to be specifically arranged. If it does, an accompanying remarkshould be included for purposes of clarity stipulating that a motionrelated solution for the problem of the trisection of an angle can beobtained only by means of properly setting such device;

RQMT 4—revealing the primary function such device is supposed toperform. Providing such details should disclose whether such proposedarticulating invention actually is sufficiently equipped to overcome theEuclidean deficiency of being unable to fully backtrack upon anyirreversible geometric construction pattern whose rendered angle is of amagnitude which amounts to exactly three times the size of its givenangle; and

RQMT 5—explaining why each device setting automatically portrays aunique motion related solution for the problem of the trisection of anangle. Providing such details should disclose whether all proposedarticulating invention device settings were substantiated individually,or incorrectly validated by means of instead applying a particular,singular solution for all cases.

When a proposed articulating invention fails to meet any, or even all,of such five above itemized requirements, it is important to note thatit still might be fully capable of performing trisection. However, itwould become virtually impossible to substantiate that such device couldperform trisection accurately throughout its entire range of devicesettings!

The detailed repercussions which would be expected to accompany suchtype of mishaps are delineated below. Therein, references are made tothe short term notations for each of such five requirements.Accordingly, if a proposed articulating invention fails to meet:

RQMT 1, then a claim as to which designated magnitudes such deviceactually would be capable of trisecting could not be made, other thanthose as specifically cited within its specification or expresslydepicted upon its accompanying drawing package;

RQMT 2, then a detailed accounting as to very manner in which suchdevice might overcome such impediment could not be furnished; asotherwise should have been reported as a capability to fully backtrackupon any irreversible geometric construction pattern, including that ofa rendered angle whose magnitude amounts to exactly three times the sizeof its given angle; thereby throwing serious doubt as to whether suchdesign contains provisions that actually enable it to surpass Euclideancapabilities;

RQMT 3, then it could become rather difficult to decipher how to operatesuch device;

RQMT 4, then it could become incredibly difficult to logically deducethat by means of properly setting such device to a designated magnitude,a static image would become regenerated wherein overlapment pointsfurthermore would become discernable that enable such designatedmagnitude to be fully backtracked upon, all the way back to itsassociated trisector; in effect, mitigating a Euclidean irreversibilitylimitation that otherwise would prevent the classical problem of thetrisection of an angle from being solved and, by overcoming suchdifficulty, thereby automatically portray a motion related solution forthe problem of the trisection of an angle; and

RQMT 5, then it could become quite difficult to fathom thatsubstantiating every unique motion related solution for the problem ofthe trisection of an angle that possibly could be automaticallyportrayed by such device would entail the generation of an entire familyof geometric construction patterns, all belonging to a specificEuclidean formulation.

Fulfilling all five requirements, as stated above, naturally would leadto a proper understanding of trisection. Such knowledge would becomeattained only after realizing that such listings actually work in tandemwith one another.

For example, by acknowledging RQMT 5 to be a true statement, it would beexpected that any newly proposed articulating trisection inventionappropriately would account for, not just one, but many individualmotion related solutions for the problem of the trisection of an angle.Hence, for any drawing which could become generated by means ofexecuting some particular Archimedes proposition, it should berecognized that it could serve to substantiate only one motion relatedsolution for the problem of the trisection of an angle. With regard tothe representative geometric construction pattern, as expressed in FIG.1B, it readily should become apparent that such singular drawing couldbe used to substantiate only one particular solution thereof. However,if such drawing instead were to become construed to be a full blownArchimedes formulation, in itself denoted by a sufficiency of Greekletter notations, along with what should be an included double arrowlocated around the outside of circular arc QS, it then would describe anentire family of geometric construction patterns, each of whichindividually would substantiate a unique motion related solution for theproblem of the trisection of an angle.

Obviously, such solutions would apply to different acute and/or obtusedesignated magnitudes, in complete accordance with those whichpreviously must have been specified in order to satisfy the provisionsof RQMT 1.

Moreover, an operating procedure, as specified in order to meet theprovisions stipulated in RQMT 3, thereafter could be thoroughly reviewedin order to verify that no considerable obstruction would preclude thesuitable trisection of an entire range of angles, as formerly indicatedin RQMT 1.

It is true that Wantzel and Galois, generally are credited as beinginstrumental in proving that an angle of designated magnitude cannot betrisected when acted upon only by a straightedge and compass.

However, what is quite intriguing about such work is that, while on theone hand relying rather heavily upon an analysis of variousprognosticated algebraic equations, on the other hand there doesn'tappear to be any tangible correlation as to how such determination, asposed in one branch of mathematics, relates to the geometric findingpresented herein that the classical problem of the trisection of anangle cannot be solved due to an availability of overlapment points;thereby cause irreversibility to occur within geometric constructionpatterns, and making it impossible to completely backtrack from renderedangles all the way back to given angles whose respective magnitudesamount to exactly one-third their size. Inasmuch as the classicalproblem of the trisection of an angle requires a Euclidean solution,accounting for why it cannot be achieved requires a geometricexplanation!

By interjecting non-geometric explanations, certain attributes thataccompany trisection difficulties most certainly can be identified, butonly at the risk of possibly perpetuating undesirable myths whichsurround such great trisection mystery; thereby preventing it from beingunlocked!

For example, consider the rather far fetched notion that the classicalproblem of the trisection of an angle actually might become solved byway of obtaining a cube root, solely by conventional Euclidean means!

Naturally, such hypothesis would discount any possibility that unity, byposing a cube root of itself, might play a key role in any of suchattempts. Nor should such cube root be confused in any way with a cubicroot that, if being a real number, instead would identify the exactlocation where a third order curve crosses the x-axis, as displayed upona Cartesian coordinate system.

Within any right triangle drawing, since the length of its hypotenuseamounts to the square root of the sum of the squares of its two sides,according to the Pythagorean Theorem, then such geometric constructionpattern would be a byproduct of addition, multiplication and square rootmathematical operations; which turned out to be the very basis ofpursuit in Wantzel's work. In connection with such premise, asconcerning the possible Euclidean extraction of a cube root, naturally aleading question which should be asked is what about cube root lengthswhose ratios with respect to a given length of unity are either rationalor quadratic irrational?

For example, the length of a straight line that is 3 inches longrepresents the cube root of another straight line that amounts to 27inches in overall length; whereby such longer straight line very easilycould be geometrically constructed simply by adding together nine ofsuch 3 inch long straight lines.

A much needed logic that seemingly appears to be grossly lacking in suchabove stated scenario is that if it incorrectly were to be acceded thatcube roots cannot be obtained solely by conventional Euclidean means,then it would have to follow that any geometric construction patternwhose rendered information, even when amounting to just a renderedlength, is of a magnitude that amounts to the cube of any portion of itsgiven data would have to be irreversible, not that it would present asolution of the classical problem of the trisection of an angle! Thisshall be further demonstrated later by means of geometricallyconstructing rendered lengths of cubed magnitudes.

In effect, Wantzel algebraically proved that addition, subtraction,multiplication, and division, as representing the various fundamentaloperations defined within number theory, could not be suitably appliedby a straightedge and compass in any combination that could solve theclassical problem of the trisection of an angle. Quite understandablysuch consideration would not apply to a geometric solution for theproblem of the trisection of an angle! For example, the mathematicaloperation of performing division by a factor of four could be achievedby conventional Euclidean means merely by bisecting a straight line, andthen bisecting each of its then separated portions again. As such, bymeans of performing such division upon the tangent of an angle whosevalue is 4/√{square root over (11)}, a new length could be obtained of1/√{square root over (11)} which would be indicative of the tangent ofits trisector. Obviously, Wantzel's non-geometric accounting, as brieflyoutlined above, couldn't possibly be expected to explain what isconsidered to be a Euclidean limitation; one which now, for the veryfirst time, is to be described as an inability to fully backtrack uponany rendered angle whose magnitude amounts to exactly three times thesize of its given angle due to an availability of overlapment points!

Whereas taking the cube root of a complex number also later on shall beshown to be synonymous with obtaining its trisector, such symbiosisrepresents yet another outstanding definition which could be attributedto trisection; but one which most certainly shouldn't be confused withany plausible explanation as to why trisection cannot be performedsolely by conventional Euclidean means!

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a depiction of the fundamental architecture of a famous Kempetrisecting device, as shown in this particular case to be set to adesignated magnitude of 143¼°; thereby divulging the whereabouts of onlythe longitudinal centerlines of linkages and radial centerlines ofinterconnecting pivot pins which collectively comprise such prior art,including those which are featured in its three strategically emplacedanti-parallelogram shaped control mechanisms used to strictly regulatethe manner in which it such device is permitted to articulate.

FIG. 1B is a prior art method for determining the trisection of anangle; except for the fact that all of the intersection points appearingtherein now are denoted by different letters. It furthermore isrepresentative of prior art, famously known as a marked rulerarrangement, in which the longitudinal centerline of a marked ruler, asdenoted therein by straight line MR includes a notch at point N, beinglocated any suitable arbitrary distance away from its tip, as located atpoint M along such longitudinal centerline; which furthermore sits atopa drawing of angle QPS, as algebraically expressed therein to be of 3θdesignated magnitude, an added circle which is drawn about such point Pat a radius which is equal in length to that of straight line segmentMN, and shows its straight line SP to be extended in such a manner thatthe longitudinal centerline of such ruler is jockeyed about so that itpasses through point Q, has its tip, M, rest somewhere upon straightline SP extended, while its notch additionally becomes located somewherealong the circumference of such drawn circle.

FIG. 1C is yet another illustration of prior art; shown to have beentruncated in order to apply specifically to trisection, as indicated byhaving its unused linkage depicted in phantom therein; which alsoexpresses the same letter designations which appear on FIG. 1B, therebymaking it easier to compare such two drawings in order to recognize thatthe fundamental architecture of such device, as represented in FIG. 1C,could be reconfigured so that it assumes the very same overall shape asthat which is depicted in FIG. 1B, thereby substantiating that, in suchparticular arrangement, such device would automatically portray a motionrelated solution for the problem of the trisection of an angle; andwhich additionally displays θ and 3θ algebraic angular notations for theexpress purpose of making it perfectly clear that such device is fullycapable of trisecting, not only a specific angle of 55° designatedmagnitude, as actually is depicted therein, but a wide range of otherdevice settings as well.

FIG. 2 is a flowchart which identifies the various elements of acomprehensive trisection methodology.

FIG. 3 is a Trisection Mystery Iteration Processes Table which itemizespertinent ramifications which are considered to underlie the very natureof a plaguing trisection mystery that has persisted for millennia.

FIG. 4 is a Figure Number Table that cites figure numbers of Euclideanformulations and drawing packages that apply to each of the fourembodiments which collective comprise such newly proposed articulatinginvention.

FIG. 5 is a Euclidean formulation, easily identified as such because itbrandishes a double arrow, as well as bears algebraic angular notationsupon it.

FIG. 6 is a derivative Euclidean formulation, as representing ageometrically constructed extension of FIG. 5.

FIG. 7 is a second derivative Euclidean formulation, therebyrepresenting a geometrically constructed extension of FIG. 6.

FIG. 8 is a third derivative Euclidean formulation, thereby representinga geometrically constructed extension of FIG. 7.

FIG. 9 is a Euclidean Formulation Rendered Angle Relation Table thatidentifies acute rendered angles which appears in each of such Euclideanformulations, as cited in FIG. 4; furthermore algebraically expressingeach of their magnitudes, as shown therein to amount to exactly threetimes the size of their respective given angles.

FIG. 10 is a CATEGORY 1, sub-classification B Conforming Aspects Chartwhich identifies similarities evident within the four constituentembodiments of such newly proposed invention, as tabulated in FIG. 4.

FIG. 11 is a Mathematics Demarcation Chart, so arranged to divulgeexactly which areas of mathematics can be represented only by a newlyproposed geometric forming process; thereby exposing where conventionalEuclidean practice actually is limited.

FIG. 12 is Trisecting Emulation Mechanism Flowchart that describes how atrisection emulation invention performs once a designated magnitudebecomes specified.

FIG. 13 is a Euclidean formulation that is representative of the famousalgebraic cubic function 4 sin³ θ−3 sin θ=sin (3θ), wherein for anymagnitude which given angle VOO′ might arbitrarily assume. respectivelengths, algebraically expressed as 4 sin³ θ and 3 sin θ, could be drawnsolely by conventional Euclidean means, such that the difference notedbetween them would equal a length that thereby could be algebraicallyexpressed as sin (3θ); in effect, enabling angle VOU′ to begeometrically constructed from such determination with its magnitudeamounting to exactly three times the size of such given angle.

FIG. 14 is a graph of three algebraic functions; wherein the functiondenoted by the top legend remains continuous within the range −1≤cosθ≤+1, the function denoted by the middle legend remains continuous forall real values of cos θ except when it is equal to zero, and thefunction denoted by the bottom legend is entirely discontinuous in thatit consists of only four discrete points, as noted within the largecircles displayed therein; whereby any continuous portions of suchcurves furthermore could be described by a virtual unlimited number ofgeometric construction patterns that belong to a particular Euclideanformulation that could be developed in much the same way as that whichis represented in FIG. 13.

FIG. 15 is a table of roots for the quartic equation 80 cos⁴ θ−4 cos³θ−60 cos² θ+6=0, along with other supporting data, as obtained byrelating the top and bottom functions denoted in such FIG. 14 legend inorder to establish the equality (4 cos³ θ−6)/(20 cos θ)=4 cos³ θ−3 cosθ.

FIG. 16 is a geometric construction pattern showing the process forgeometrically solving parabolic equations of the form ax²+bx+c=0; merelyby means of applying such famous Quadratic Formula x=(−b±√{square rootover (b²−4ac)})/2a solely by conventional Euclidean means for thespecific case when the coefficients a=−2, b=0.4, and c=0.75.

FIG. 17 is a geometric solution for the problem of the trisection of anangle whose designated magnitude is algebraically expressed as 3θ andwhose tangent, denoted as ζ, is assigned a value of √{square root over(5)}/7. Although not representing a bona fide solution for the classicalproblem of the trisection of an angle, which cannot be solved, suchgeometric solution does succeed at resolving a quadratic equation thatassumes the algebraic form z_(R) ²+b′z_(R)+c′=0, as obtained by means ofapplying a particular abbreviated version of the Quadratic Formulaz_(R)=(½)(−b′±√{square root over (b′²−4c′)}) to it for the particularcase when b′=(3+γ)/(3ζ+β) and c′=(δ−ζ)/(3ζ+β), thus amounting tob′=−(105+49√{square root over (5)}))/(4√{square root over (5)}+49) andc′=(85√{square root over (5)})/(49+4√{square root over (5)}) for theparticular quadratic equation which results when two cubic equations ofa singular variable known to share a common root expressed, z_(R)=tan θ,become simultaneously reduced, solely in algebraic fashion, when each isrepresented as:

ζ=tan(3θ)=√{square root over (5)}/7=(3z _(R) −z _(R) ³)/(1−3z _(R)²);and

z _(R) ³ +βz _(R) ² +γz _(R)+δ=0 when β=−(√{square root over(5)}+7),γ=7√{square root over (5)}+12, and δ=−12√{square root over (5)}.

FIG. 18 is a diagram that indicates how an angle of arbitrarily selecteddesignated magnitude, denoted algebraically as 3θ therein, can betrisected by means of geometrically constructing a series of properlyarranged successive Euclidean bisections.

FIG. 19 is a Successive Bisection Convergence Chart that discloses themeasure of trisection accuracy which could be obtained by means ofincreasing the number of properly arranged successive Euclideanbisections that take place within a particular geometric constructionprocess; thereby indicating that just after twenty-one iterations, asindicated in the line item in which n=22 therein, trisection would beperformed to an accuracy of six decimal places if the human eye werecapable of detecting such activity.

FIG. 20 is a diagram of a complex number whose arbitrarily selectedangular magnitude, algebraically denoted as θ therein, serves both as atrisector for, as well as a cube root of another complex number thatbecomes geometrically constructed with respect to it such that itsmagnitude amounts to exactly three times its size, thereby beingalgebraically designated as 3θ therein.

DETAILED DESCRIPTION

Certainly by now it should have been made quite clear that in order tounlock vital secrets, highly suspected to be hidden deep within the veryrecesses of a perplexing trisection mystery, a paradigm shift mostdefinitely is warranted; one that expressly should recommend somefundamental change in overall approach concerning how to properlyaccount for difficulties encountered when trying to solve the classicalproblem of the trisection of an angle.

Only by means of exposing such closely held secrets could the basicobjective of a comprehensive trisection methodology become realized, aspresented in the flowchart appearing in FIG. 2; essentially being tovalidate that the design of some proposed invention could performtrisection accurately throughout a wide range of device settings and, inso doing, qualify as a legitimate trisecting emulation mechanism thatcan automatically portray various motion related solutions for thetrisection of an angle.

Accordingly, a detailed discussion of such flowchart should precede theintroduction of such newly proposed invention. In this way, anyrequirements posed relating to the design of its four constituentembodiments would be presented well before explaining exactly how theyare to complied with. Such accounting begins with a process box entitledMATHEMATIC LIMITATION IDENTIFIED 1 therein, representing the task withinsuch flowchart where some unknown mathematical limitation is identifiedthat supposedly prevents the classical problem of the trisection of anangle from being solved. Obviously, since such solution must dependsolely upon the communication of a straightedge and compass with respectto an angle of designated magnitude, any mathematic limitation alludedto therein must be some pronounced difficulty having to do withconventional Euclidean practice!

The process box referred to as UNKNOWN GEOMETRIC PROPERTY UNCOVERED 2 iswhere, in the course of such FIG. 2 flowchart, an entirely new geometricproperty is to be uncovered which furthermore is considered to be thecause of such identified mathematic limitation. Although presently beingunknown, any newly defined geometric property naturally would have to beas basic a shape as a well known straight line or circle; thereby makingsuch trisection mystery that much more intriguing.

The third process box, entitled DEGREE OF IMPOSITION DELINEATED 3 isreserved for describing the extent of difficulty that such newlyuncovered geometric property is anticipated to impose upon conventionalEuclidean practice.

The process box referred to as DEVICE PRIMARY FUNCTION REVEALED 4 iswhere an as yet unknown capability thereby becomes revealed whichassumes the form of some specially added equipment that articulatingmechanisms can be fitted with that enables them to overcome, correct, orcompensate for such undermining influence, as now suspected to be amathematic limitation.

Next, the decision box entitled DEFICIENCY MITIGATED 5 within such FIG.2 flowchart serves to verify that certain equipment featured in suchproposed articulating devices that are supposed to avail such suspectedprimary function actually are deemed to be of sufficient designs tosuitably mitigate such adverse influence. If it turns out that they arenot adequate to perform such identified primary function, then theyrequire redesign. If, instead, it turns out that they perform suchprimary function, but do not trisect, then such suspected mathematiclimitation must be an incorrect selection, and another response therebyneeds to be chosen. The recourse for such noted action is indicated bythe NO pathway which is shown to exit such decision box.

The input box entitled TRISECTION RATIONALE 6, as shown in FIG. 2, iswhere a discussion is presented that accounts for how the correctresponses, as indicated in such FIG. 3 Trisection Mystery IterationProcesses Table, were chosen in the very first place.

Such trisection rationale discussion specifically directs attention tothe first four processes listed in such FIG. 2 flowchart, and proceedsby conjecturing that overlapment points residing within an irreversiblegeometric construction pattern elude detection from any and allEuclidean interrogations which possibly could be launched exclusivelyfrom the sole vantage point of its rendered information.

The very fact that overlapment points remain entirely inconspicuous inthis manner furthermore evidences that it is impossible to specify adistinct set of Euclidean commands which can identify their whereaboutssolely with respect to such rendered information.

Without such vital input, a specific sequence of Euclidean operationsfurthermore could not be developed that instructs how to apply astraightedge and compass in order to trace out a pathway which begins atsuch rendered information and leads all the way back to a given set ofpreviously defined geometric data; whereby the very presence ofoverlapment points serves to circumvent reversibility!

Since the very concept of reversibility is entirely new with regards toconventional Euclidean practice, a validation that isosceles triangleMNP, as posed in FIG. 1B, is a reversible geometric construction patternis afforded directly below:

whereas the first three steps of a previously stipulated sequence ofEuclidean operations already has accounted for how to geometricallyconstruct isosceles triangle MNP directly from given acute angle RMP,all that is needed in order to demonstrate reversibility is to therebygeometric construct isosceles triangle MNP with respect to its renderedangle PNM instead, as is outlined in the three step sequence ofEuclidean operations which follows:

step 1—from rendered ∠PNM, an arbitrary length NM is marked off alongone of the sides with point M becoming assigned to its newly describedend;

step 2—a circular arc is swung about point N whose radius is of lengthNM; and

step 3—point P becomes designated at the newly determined intersectionof such circular arc with the other side of ∠PNM, whereby straight linesNP and PM become drawn to complete isosceles triangle MNP.

In order to demonstrate the actual difficulty which an intrusion ofoverlapment points causes, notice in FIG. 1B that it is impossible togeometrically construct isosceles triangle MNP solely with respect torendered angle QPS.

Taking any of the specific geometric construction patterns whichcollectively constitute such Archimedes formulation into account, thisbecomes evident upon realizing that overlapment points M and N, asrepresented in such FIG. 1B, never could be located solely with respectto rendered angle QPS by conventional Euclidean means. The reason forsuch impossibility is furnished below:

even though it is known that overlapment point M must reside somewherealong straight line SP extended, it cannot be determined solely viastraightedge and compass exactly which of the infinite number ofpossible locations which resides upon it applies when commencingexclusively from rendered angle QPS; and

the same argument holds true for overlapment point N which is known toreside somewhere along a circle that is drawn about point P that is ofradius PQ, but whose exact location cannot be precisely pinpointedexclusively with respect to rendered angle QPS solely via straightedgeand compass.

For the particular hypothetical case when QPS amounts to exactly ninetydegrees, such thirty degree trisector very easily could be geometricallyconstructed, simply by bisecting any angle or side of an equilateraltriangle. However, the computation of dividing such ninety degree angleby a factor of three in order to arrive at the magnitude of such thirtydegree trisector unfortunately cannot be duplicated solely byconventional Euclidean means. Hence, to do so only would create acorrupted version of the classical problem of the trisection of anangle; thereby solving an entirely different problem!

Hence, in such capacity, overlapment points function as obstructionsserving to confound attempts to redefine an entire geometricconstruction pattern solely with respect to its rendered information.

Consequently, any pathway consisting of previously distinguishedintersection points which originally led from given angle RMP all theway to rendered angle QPS, as depicted in FIG. 1B, could not be retracedin complete reverse order by means of attempting to apply only astraightedge and compass with respect to such rendered angle QPS.

In that such discussion particularly should account for difficultiesexperienced when attempting to solve the classical problem of thetrisection of an angle, it thereby becomes formally stipulated that itis impossible to fully backtrack upon any geometric construction patternwhose rendered angle is of a magnitude that amounts to exactly threetimes the size of its given angle; simply because such drawing wouldharbor overlapment points!

As such, a presence of overlapment points within such specific types ofgeometric construction patterns entirely thwarts attempts to generatesuch overall pathways in complete reverse order, solely by conventionalEuclidean means; thereby preventing the classical problem of thetrisection of an angle from being solved!

In summary, overlapment points have an affinity to impede the completionof geometric construction patterns that are replete with them for themere reason that they cannot be entirely reconstituted solely viastraightedge and compass in complete reverse order.

For the benefit of any remaining skeptics, it furthermore should beadded that only when the magnitude of a trisected angle becomesfurnished beforehand can a geometric construction pattern whichspecifies such trisector, in the very the form of its given angle,become fully reversible; thereby enabling some corrupted version of theclassical problem of the trisection of an angle to be solved.

During such condition, overlapment points, by definition, then wouldbecome distinguishable intersection points with respect to such giventrisecting angle; thereby making such geometric construction patternfully reversible. However, to attempt such activity would defeat thepurpose of trying to trisect an angle solely by conventional Euclideanmeans in the very first place; simply because the very information beingsought after already has been furnished. In other words, it would beentirely senseless to generate geometric quantities such as straightlines, circles, and angles aforehand exclusively for purposes of thendetermining them solely via straightedge and compass. Nevertheless, anotable history of this exists which mostly has been directed towardsimproper attempts to trisect angles solely via straightedge and compass.

Such foolish endeavors stand is sharp contrast to most, if not all,other standard Euclidean procedures, such as bisection; whereby abisector remains totally unknown until such time that it actuallybecomes geometrically constructed from an angle of given magnitude.

When only the magnitude of an angle that is intended to be trisectedbecomes designated, its associated geometric construction patternremains completely unspecified. This presents a heightened problembecause there virtually are a countless number of other geometricconstruction patterns, besides those represented in FIGS. 1A and 1B,that also render angles whose magnitudes amount to exactly three timesthe size of respective given angles. Without being informed as to whichparticular geometric construction pattern applies in the very firstplace, resident overlapment points no longer become limited to specificintersection point locations upon a specific pattern.

Even when a specific geometric construction pattern becomes selected asa vehicle for attempting to perform trisection, such as in the case ofthe rendition of the Archimedes formulation, as posed in FIG. 1B, itsgiven angle NMP or RMP, even when designated to be of a specified sizethat can be duplicated solely by means of applying a straightedge andcompass, still cannot be determined when launching Euclidean operationsjust with respect to its rendered angle QPS; principally because itsresident overlapment points cannot be distinguished.

Such pronounced geometric construction limitation of not being able toencroach upon overlapment points when being launched from a particulardirection can, in fact, be rectified rather simply; merely by affordinga means for discerning overlapment points that reside withinirreversible geometric construction patterns, and thereby making thementirely distinguishable with respect to rendered angles which otherwisecannot be backtracked upon!

Such elementary recommendation, despite its rather unsuspecting andseemingly outlandish nature, nevertheless describes exactly how atrisecting emulation mechanism can trisect virtually any designatedangle which it can be set to; thereby portraying a of motion relatedsolution for the problem of the trisection of an angle.

Such strange phenomena perhaps most easily can be described with respectto the motion of any CATEGORY I sub-classification A articulatingtrisection device because such types of devices do not first have to bespecifically arranged before displaying their settings. As any of suchdevices becomes cycled, eventually reaching all of the settings withinits entire operating range, its fundamental architecture sweeps out, orregenerates, a multitude of static images, each representing a stillshot cameo of two angles, the larger of which not only amounts toexactly three times the size of the other, but furthermore is calibratedto a specific device setting.

The beauty of such design concept is that once any of such types ofdevices becomes set to a preselected designated angle, the portion ofthe smaller angle contained within the static image which becomesregenerated thereby automatically portrays its associated trisector.

In other words, by means of properly setting any trisecting emulationmechanism, its fundamental architecture becomes rearranged to aparticular position such that the static image which becomes regeneratedautomatically portrays a motion related solution for the problem of thetrisection of an angle!

In effect, such motion related solution distinguishes overlapment pointswhose availability otherwise would prevent the classical problem of thetrisection of an angle from being solved!

Accordingly, instead of attempting to perform that which is impossible;essentially consisting of retracing a distinguishable pathway within anirreversible geometric construction pattern in complete reverse ordersolely by conventional Euclidean means, a trisecting emulation mechanismotherwise functions like the Dewey decimal system in a library whereinthe exact name of a document that is being searched for becomes eitherinput into a computer, or otherwise looked up in some card deck, wherebyan alpha-numeric code that provides an indication of its whereabouts,thereby allows such information to forthwith become retrieved. The onlyglaring difference in the case of a trisecting emulation mechanism isthat the magnitude of a designated angle which is slated for trisectionbecomes set into such device, thereby causing the regeneration of aparticular static image that automatically portrays its associatedtrisector!

Accordingly, a fundamental architecture might be thought of as amechanical means for conveniently storing a multitude of static imageswithin the very memory of some particularly designed trisectingemulation mechanism; thereby enabling a motion related solution for theproblem of the trisection of an angle of designated magnitude to beautomatically portrayed at will.

To conclude, a unique pathway which leads from one angle all the way toanother that amounts to exactly three times its size automaticallybecomes portrayed each and every time a static image become regeneratedby means of configuring a trisecting emulation mechanism to any of itsdiscrete device settings; thereby disclosing the actual whereabouts ofnuisance overlapment points which reside along the way; simply by meansof exposing them to be nothing more than commonly known intersectionpoints. In so doing, any obstructions that otherwise normally would beencountered when attempting to solve the classical problem of thetrisection of an angle, would be overcome merely by means of properlysetting a trisecting emulating mechanism.

A basic tenet of conventional Euclidean practice is that all activitymust proceed exclusively from a given set of previously definedgeometric data, or else from intersection points which become locatedwith respect to it.

It may well be that a purposeful adherence to such rule might explainwhy any serious attempt to completely retrace a geometric constructionpattern exclusively from its rendered information all the way back toits given set of previously defined geometric data, solely byapplication of a straightedge and compass, entirely might have beenoverlooked in the past.

Moreover, only on very rare occasions, such as in the particular case ofattempting to solve the classical problem of the trisection of an angle,could the prospect of possibly even engaging upon such activity arise,thereafter culminating in an avid interest to solve such classicalproblem without considering that a pathway leading from a rendered anglewithin any geometric construction pattern all the way back to a givenangle whose magnitude amounts to exactly one-third of its size lies atthe very heart of such difficulty!

Remarkably, only by means of analyzing conventional Euclidean practicefrom this other seldom viewed perspective could irreversibility beidentified as being caused by an intrusion of overlapment points.

By otherwise neglecting such critical information, it would becomevirtually impossible to substantiate that any qualifying CATEGORY Isub-classification A or CATEGORY II articulating trisection mechanismcould perform trisection accurately throughout a wide range of devicesettings.

The input box entitled IMPROVED DRAWING PRETEXT 7, as posed FIG. 2, iswhere a new truncated drawing format is to be introduced that canrepresent an entire family of geometric construction patterns, all uponjust a single piece of paper.

Whereas such FIG. 2 flowchart is meant to apply exclusively totrisection, such improved drawing pretext, as alluded to therein,preferably should be identified as any Euclidean formulation each ofwhose constituent geometric construction patterns depicts a renderedangle whose magnitude amounts to exactly three times the size of itsgiven angle. By means of suitably designing a trisecting emulationmechanism so that virtually any static image which would becomeregenerated as the result of its being properly set thereby wouldautomatically portray an overall outline that furthermore could be fullydescribed by a particular geometric construction pattern which belongsto such Euclidean formulation, then it could be substantiated that amotion related solution for the problem of the trisection of a anglecould be achieved, merely by means of backtracking upon an irreversiblecondition that instead would have prevented the classical problem of thetrisection of an angle from being solved!

Accordingly, the rather seemingly antiquated idea of generatingsingular, but unrelated geometric construction patterns thereby veryeasily could become dwarfed simply by means of considering the prospectthat they furthermore might become linked to one another in someparticular fashion through the use of an improved drawing pretext forthe express purpose of geometrically describing motion!

The wording above is intended to infer that improved drawing pretexts,other than that of the Euclidean formulation could be devised, therebyassociating their constituent drawing patterns in some distinct mannerother than through specified sequences of Euclidean operations; and,upon becoming replicated might thereby describe important motions whichare known to be of service to mankind!

Such discussion is building to the proposition that by means of properlypartitioning all observed phenomena which can be describedgeometrically, including that of certain motions, it thereby becomespossible to envision a certain order that becomes evident within afarther reaching mathematics.

Such is the very purpose of the input box entitled MATHEMATICSDEMARCATION 8, as posed in FIG. 2 herein. Its key artifact consists of aMathematics Demarcation Chart, as posed in FIG. 11, which discloses aparticular partitioning which should be imposed universally in order tosuitably distinguish between geometries which describe stationarypatterns, as opposed to those which can quantify disparate motionrelated geometries.

As it pertains to trisection matters, the drawing pretext entryappearing in the third column of such FIG. 11 chart, as listed directlyunder the cell entitled Geometric forming process, quite expectedly,turns out to be that of a Euclidean formulation; hence, limiting overallscope therein to matters in which geometric construction patterns can beassociated to one another only through particular sequences of Euclideanoperations.

Headings appearing in FIG. 11, are shown to run along the left side ofsuch chart. Such arrangement enables the two principal listingsappearing at the top of the second and third columns therein to serve asminor headings in themselves; thereby making it easy to differentiatebetween conventional Euclidean practice and a geometric forming processmerely by means of comparing such two columns to one another.

Moreover, inasmuch as the field of geometry concerns itself withmathematically quantified depictions, algebra, on the other hand, byrepresenting the overall language of mathematics, instead bears thebiggest brunt of responsibility in validating that such alleged ordertruly exists; doing so by associating algebraic format types throughsome newly proposed equation sub-element theory!

One principal reference, standing as a harbinger of a newly proposedequation sub-element theory, is a relatively unknown treatise that waspublished in 1684; as written by one Thomas Baker and entitled, TheGeometrical Key or the Gate of Equations Unlocked. After a closeaffiliation with Oxford University, Mr. Baker successfully provided asolution set pertaining to biquadratic equations, perhaps more commonlyreferred to today as either quartic, or fourth order equations. However,it seems quite plausible that because of a serious competition amongrival institutions going all the way back to that time period, GerolamoCardano's preceding work of 1545, as it appeared in Ars Magna,nevertheless, still managed to eclipse his later contributions. Inbrief, Cardano applied a transform to remove the second, or squared,term from cubic equations in order to modify them into an overall formatthat very easily could be resolved. However, because of such grosssimplification, the all important fact that each algebraic equation isunique, in its own right, was largely ignored; hence, failing toattribute deliberate meaning to the various equation types that actuallygovern third order algebraic equation formats. The very stigma whichsuch abbreviated process instilled unfortunately served to directattention away from developing an all purpose solution that applies toall cubic equation formats, as posed in a single variable; one whichobviously would lie at the very heart of any newly proposed sub-elementtheory; thereby not requiring that cubic equations which express secondterms first become transformed in order to solve them! In retrospect, itnow appears very likely, indeed, that a hit-and-miss mathematicsapproach of such nature most probably delayed the actual debut of anewly proposed equation sub-element theory by some four hundred years!

To conclude, by means of now introducing an all-purpose cubic equationsolution, as presently has remained absent for all these years, the veryrelevancy of each format type can remain preserved so that furthercomparisons could be made in order to avail a more comprehensiveunderstanding of an overall order that actually prevails within all ofmathematics.

In such FIG. 11 Mathematics Demarcation Chart, notice that cubicirrational numbers are listed only under the heading referred to asgeometric forming process. Therein, such partitioning assignment isentirely consistent with the proposed finding that although angularportions within a regenerated static image can become automaticallyportrayed, even when they consist of cubic irrational trigonometricproperties, nevertheless such angles cannot be geometric constructedjust from a given length of unity or from another angle whosetrigonometric properties are either rational or quadratic irrational!

That is to say, whenever the angular portion within a regenerated staticimage that has been calibrated to a particular device setting bearscubic irrational trigonometric properties, so must the angular portiontherein which serves as its trisector. Accordingly, there is no way torelate either rational or quadratic irrational trigonometric propertiesof a trisector to an angle which amounts to exactly three times its sizethat bears cubic irrational trigonometric properties.

In other words, it requires, not one, but three angles that all exhibitcubic irrational trigonometric properties in order to geometricallyconstruct an angle which exhibits either rational or quadraticirrational trigonometric properties. Such angle very well could begeometrically constructed in a manner which is analogous, or consistentwith virtually any of the nine the arrangements of such products, sums,and sums of paired products, as posed in the algebraic equationspreviously expressed in such definition of a cubic irrational number.

Accordingly, any geometric construction pattern that belongs to aEuclidean formulation which furthermore is known to replicate thearticulated motion of the fundamental architecture of any CATEGORY Isub-classification A trisecting emulation mechanism which therebybecomes reset every time it becomes articulated only can be approximatedin size if it is meant to depict a static image either of whose twoincluded angular portions portrays cubic irrational trigonometricproperties!

An elementary, but nonetheless very revealing example of this concernsattempts to trisect a sixty degree angle solely by conventionalEuclidean means!

Although such sixty angle can be distinguished merely by geometricallyconstructing an equilateral triangle, its associated twenty degreetrisector, on the other hand, is known to exhibit transcendentaltrigonometric properties that cannot be geometrically constructed, whenproceeding either exclusively from a given length of unity, or solelyfrom any angle whose trigonometric properties exhibit either rational orquadratic irrational values.

Such explicitly stated impossibility is what actually distinguishes therealm between where angles can be portrayed which bear cubic irrationaltrigonometric property values, and other angles that do not whichthereby can be expressed solely by conventional Euclidean means!

Further note in such FIG. 11 chart that linear, as well as quadraticalgebraic equation and associated function format type entries appearunder both conventional Euclidean practice, as well as geometric formingprocess cells. This is because linear straight lines and/or second ordercircular arcs which remain stationary over time amidst an agitatedmotion would assume the very same shapes within each and every geometricconstruction pattern which belongs to any Euclidean formulation thatfurthermore could be animated in order to replicate such articulationevent; thereby applying to both sides of such partitioned FIG. 11 chart.

The fact that cubic equations appear only under the heading referred toas geometric forming process therein is a little more difficult toexplain; having to do with the fact that by depicting actual motions,Euclidean formulations moreover can be expressed algebraically ascontinuums.

The most commonly known algebraic continuum is an infinite series whoseterms become summed over some specific predetermined range ofperformance.

It naturally follows then that their integral counterparts, as realizedwithin the field of calculus, also could apply, as well, to certainrelative motions which furthermore can be geometrically described byEuclidean formulations. Quite obviously, this presumption moreoverassumes that such motions actually do appear as complete continuums toany would be observer, wherein the time interval pertaining to suchintegral sign would approach zero; thereby confirming the very validityof yet another rather intrusive mathematical involvement.

Furthermore, other types of algebraic equations are considered to becontinuous, beginning with that of a straight line whose linear equationof y=mx+b validates that for each and every real number x which becomesspecified, a corresponding value of y truly exists.

With particular regard to a motion related solution for the problem ofthe trisection of an angle, algebraically expressed continuums relate toEuclidean formulations by well known cubic equations of a singlevariable in which trigonometric values of an angle of size 3θ becomeassociated to those of an angle of size θ.

The key factor pertaining to such relationships is that no matter whatvalues might be applied to either of such angles, a three-to-onecorrespondence nevertheless would hold between their respective angularamplitudes!

As an example of this, consider various motion related solutions for theproblem of the trisection of an angle which could be portrayed whencycling such famous Kempe prior art from a 20 degree setting to one of120 degrees.

In such case, not only would an entire Euclidean formulation withrepresentative geometric construction pattern as fully described by FIG.1A geometrically describe such three-to-one angular correspondenceduring device flexure, but so too would the well known algebraic cubicfunction which assumes the form cos (3θ)=4 cos³ θ−3 cos θ.

That is to say, within such Euclidean formulation, angle ABC, whenamounting to virtually any designated magnitude 3θ within the limits of20°≤∠ABC≤120°, furthermore would algebraically relate to an angle ABDtherein, of resulting size θ, by such aforementioned famous algebraiccubic function.

Algebraically, such relationship could be confirmed for virtually anyangle within such postulated range. For example, below such functionalrelationship is confirmed algebraically for the particular conditionwhen angle ABC amounts to exactly 60°:

∠ ABC = 3θ = 60^(′) θ = 60^(′)/3 = 20^(∘) = ∠ ABD;cos   (∠ ABC) = cos   (3θ) = cos   60^(∘) = 0.5;cos   (∠ ABD) = cos   θ = cos   20^(∘) = 0.93969262  …  ;$\begin{matrix}{{{4\mspace{14mu} \cos^{3}\; \theta} - {3\mspace{14mu} \cos \mspace{14mu} \theta}} = {{4\left( {{0.9}3969262\mspace{14mu} \ldots}\mspace{14mu} \right)^{3}} - {3\left( {{0.9}3969262\mspace{14mu} \ldots}\mspace{14mu} \right)}}} \\{= {{3.319077862} - {{2.8}19077862}}} \\{= {0.5.}}\end{matrix}$

Additionally, a specific nature that is found to be evident withinalgebraic continuums furthermore shall become addressed, wherein:

a Euclidean formulation, each of whose constituent geometricconstruction patterns exhibits a rendered angle whose magnitude amountsto exactly three times the size of its given angle, is to becomeobtained by means of having the value of its sine described by a lengthof 3 sin θ−4 sin³ θ; thereby conforming to the famous cubic function 3sin θ−4 sin³ θ=sin (3θ); and

a graph is to become developed that distinguishes between the continuityof such well known cubic function 4 cos³ θ−3 cos θ=cos (3θ) and thediscontinuity that clearly is evident within a function that otherwiseassumes the form (4 cos³ θ−6)/(20 cos θ)=cos (3θ).

Note that in this presentation such issue is addressed even before amore important detailed discussion that shall describe the very designsof such four newly proposed embodiments.

One method of algebraically relating a quadratic equation to twoindependent cubic functions that share a common root, wherein eachfunction is limited only to a singular variable, is to link theirrespective coefficients together by means of what commonly is referredto as a simultaneous reduction process.

Since such common root, as denoted as z_(R) below, occurs only when thevalue y in such functions equals zero, the following second orderparabolic equation, thereby assuming the well known form ax²+bx+c=0, canbe derived from the following two given cubic equations:

y ₁=0=z _(R) ³+β₁ z _(R) ²+γ₁ z _(R)+δ₁;

y ₂=0=z _(R) ³+β₂ z _(R) ²+γ₂ z _(R)+δ₂;

z _(R) ³+β₁ z _(R) ²+γ₁ z _(R)+δ₁=0=z _(R) ³+β₂ z _(R) ²+γ₂ z _(R)+δ₂;

β₁ z _(R) ²+γ₁ z _(R)+δ₁=0=β₂ z _(R) ²+γ₂ z _(R)+δ₂;

0=(β₂−β₁)z _(R) ²+(γ₂−γ₁)z _(R)+(δ₂−δ₁); and

0=az _(R) ² +bz _(R) +c.

Therein, whenever coefficients a, b, and c become specified, a straightline of length equal to such common root z_(R) can be determined solelyby conventional Euclidean means, simply by developing a geometricconstruction pattern that is representative of the famous QuadraticFormula z_(R)=(−b±√{square root over (b²−4ac)})/2a. Since such approachis not germane just to trisection, but nevertheless is relevant to aproper understanding of the dichotomy which exists between cubicfunctions of a single variable and an algebraically related famousparabolic equation, such geometric construction approach is to bedescribed later on; after the four embodiments of such newly proposedinvention first become formally introduced. Moreover, such particularresolution shall pertain to the specific circumstance when thecoefficients in such well known parabolic equation, assuming theparticular form az_(R) ²+bz_(R)+C=0=ax²+bx+c become assigned therespective values of a=−2, b=0.4, and c=0.75, thereby later beingdescribed by the second order equation of a single variable of theparticular form −0.2x²+0.4x+0.75=0.

In such FIG. 11 Mathematics Demarcation Chart, algebraic equations andtheir associated functions are addressed interchangeably. Suchassociation between them easily can be recognized when considering thatby reformatting the function stipulated above into equation format, itsoverall content in no way changes, but only becomes perceived from acompletely different perspective, such that:

z ³ +βz ² +γz+δ=y; and

z ³ +βz ² +γz+(δ−y)=0

In such first case, the variable z can change in value, therebypromoting a new corresponding value for y.

However, in such second case, generally a specific value of z is beingsought after based upon the particular values which are assigned to itssecond order coefficient β, its linear coefficient γ, and its scalarcoefficient δ−y. Notice that in such particular later reformatting, noattention whatsoever is directed to the fact that such value y alsosignifies a particular height above an x-axis within an orthogonalcoordinate system at which a horizontal line passes through the curvethat can be algebraically expressed as z³+z²+γz+δ=y at three specificlocations whose corresponding values away from the y-axis amount to therespective magnitudes of z. Such perceived distinctions also suitablyshould be accounted for, in order to serve as yet other rudimentaryelements, as contained within an all-encompassing newly proposedequation sub-element theory.

In such FIG. 11 chart, it further is indicated that only certain realnumbers can reside within specific algebraic equations types, as well astheir associated functions; thereby even further evidencing an overallorder that exists within a farther reaching mathematics!

Such relationships are further addressed in section 9.3, as entitledCubic Equation Uniqueness Theorem, also appearing within such abovecited treatise; wherein it is stated that with respect to equationformats of singular variable, “Only cubic equations allow solelyrational and quadratic irrational numerical coefficients to co-existwith root sets comprised of cubic irrational numbers”.

Such technical position doesn't address higher order equations merelybecause they represent byproducts of cubic relationships which arefashioned in a singular variable.

Neither does such contention dispute, nor contradict the fact that cubicirrational root pairs can, and do exist within quadratic equations ofsingular unknown quantity.

An example of this follows with respect to the parabolic equationpresented below, followed by an associated abbreviated form of theQuadratic Formula:

ax² + bx + c = 0 ${x^{2} + {\frac{b}{a}x} + \frac{c}{a}} = 0$x² + b^(′)x + c^(′) = 0; and $\begin{matrix}{x = \frac{{- b} \pm \sqrt{b^{2} - {4\; {ac}}}}{2a}} \\{= \frac{{{- b}/a} \pm {\left( {1/a} \right)\sqrt{b^{2} - {4\; {ac}}}}}{2{a/a}}} \\{= \frac{{{- b}/a} \pm \sqrt{{b^{2}/a^{2}} - {4\; {{ac}/a^{2}}}}}{2}} \\{= \frac{{{- b}/a} \pm \sqrt{\left( {b/a} \right)^{2} - {4{c/a}}}}{2}} \\{= {\frac{{- b^{\prime}} \pm \sqrt{b^{\prime 2} - {4\; c^{\prime}}}}{2}.}}\end{matrix}$

After examining such abbreviated Quadratic Formula, it becomes obviousthat the only way in which such roots can be of cubic irrational valueis when either coefficient b′ and/or c′ also turns out to be cubicirrational.

As such, a corollary furthermore states, “Cubic irrational root pairswhich appear in parabolic equations or their associated functionsrequire supporting cubic irrational coefficients”.

Just as in the general case of conventional Euclidean practice wherestringent rules apply, so to should they be specified in support of ageometric forming process. With respect to such flowchart, as posed inFIG. 2, such entries pertain to the input box entitled SET OF RULES 9.

A few of the very simple rules which apply to geometric forming areelicited directly below. Their intent is to simplify the overalladministration of such process by means of requiring fewer lines in anyattendant substantiation. As duly furnished below, some of them mightappear to be rather straightforward, even to the point where they may beconsidered as being somewhat obvious such that:

one principal rule is that the overall length of a linkage which belongsto any trisecting emulation mechanism is considered to remain constantthroughout device flexure. Naturally, such rule applies so long as thelinkage under consideration remains totally inelastic and intact duringdevice flexure. From such rule, a wide variety of relationships therebycan be obtained, a small portion of which are listed as follows:

when two straight solid linkages of equal length become attached alongtheir longitudinal centerlines at a common end by an interconnectingpivot pin which situated orthogonal to it, such three piece assemblythereby shall function as an integral hinged unit, even duringconditions when one of such linkages becomes rotated respect to theother about the radial centerline of such interconnecting pivot pin; and

whenever one free end of such integral three piece unit thereby becomesattached along its longitudinal centerline to the solid end of anotherstraight slotted linkage along its longitudinal centerline by means ofinserting an second interconnecting pivot pin through a common axiswhich is orthogonal to such longitudinal centerlines, and thereafter theremaining unattached end of such initial integral three piece hingedunit has a third interconnecting pivot pin inserted orthogonally throughits longitudinal axis whose radial centerline lies equidistant away fromthe radial centerline of its hinge as does the radial centerline of suchadded second interconnecting pivot pin, whereby such thirdinterconnecting pivot pin furthermore passes through the slot of suchslotted linkage, the longitudinal centerlines of such three linkages,together with the radial centerlines of such three interconnecting pivotpins collectively shall describe an isosceles triangle shape in space,even during device flexure. For example, when viewing prior art, asposed in FIG. 1C, notice that the triangle whose vertices are describedby axis M, axis N, and axis P must remain isosceles no matter whatmagnitude becomes applied to angle RMS. Such is the case because theconstant distance between axis N and axis M always must be equal to thatwhich lies between axis N and axis P therein;

a second rule which more particularly pertains to trisection is that thevarious shapes that collectively comprise an entire family of geometricconstruction patterns all bear a distinct geometric relationship to oneanother based upon the fact that they all stem from the very samesequence of Euclidean operations. A few examples of how such rule can beadministered are presented below:

the radial centerlines of interconnecting pivot pins which becomeconstrained within trisecting emulation mechanism linkage slots mustremain aligned along the straight line, or even curved paths of theirrespective longitudinal centerlines during device flexure; as based uponthe design principle that the constant width of such slot, whoselongitudinal centerline also remains coincidental with that of suchslotted linkage, is just slightly larger than the diameter of the shanksof the interconnecting pins which are constrained within it;

static images which become regenerated whenever a CATEGORY Isub-classification A trisecting emulation mechanism becomes cycled overits wide range of device settings automatically portray a virtuallyunlimited number of unique overall shapes which furthermore fully can befully described by a Euclidean formulation; more particularly meaningthat for any discrete device setting, the longitudinal centerlines oflinkages and radial centerlines of interconnecting pivot pins whichconstitute its fundamental architecture furthermore can be described bythe respective straight lines and intersection points of a geometricconstruction pattern which belongs to such Euclidean formulation; and

as a CATEGORY I sub-classification A trisecting emulation mechanismbecomes cycled over a wide range of device settings, any change whichcan be observed in the magnitude of the intrinsic angles of itsfundamental architecture furthermore fully can be described by thosewhich become exhibited between corresponding straight lines within aEuclidean formulation which describes the overall shapes of itsportrayed static images;

another rule is that both rational, as well as quadratic irrationalnumbers can be algebraically equated to specific sets of cubicirrational numbers. The procedure for accomplishing this consists offirst selecting a specific rational or quadratic irrational number thatis to be characterized and then setting it equal to the left-hand sideof one of nine equations presented in the preceding definition of acubic irrational number, wherein:

for the particular case when a rational number of ⅛ is to be furthercharacterized, the first of such nine equations can be applied in orderto determine a value for 3θ₁ as follows wherein each concluding threedot notation indicates that such number extends an infinite number ofdecimal places to the right, thereby being indicative of an actual cubicirrational number:

cos (3 θ₁)/4 = 1/8 = cos   θ₁  cos   θ₂  cos   θ₃;cos (3θ₁) = 1/2 $\begin{matrix}{{3\theta_{1}} = {{arc}\mspace{14mu} {cosine}\mspace{14mu} {1/2}}} \\{{= {60{^\circ}}};}\end{matrix}$ $\begin{matrix}{\theta_{1} = {60{{^\circ}/3}}} \\{= {20{^\circ}}}\end{matrix}$ $\begin{matrix}{{\cos \mspace{14mu} \theta_{1}} = {\cos \mspace{14mu} 20{^\circ}}} \\{{= {0.93969262\mspace{14mu} \ldots}}\mspace{14mu};}\end{matrix}$ $\begin{matrix}{\theta_{2} = {\theta_{1} + {120{^\circ}}}} \\{= {{20{^\circ}} + {120{^\circ}}}} \\{{= {140{^\circ}}};}\end{matrix}$ $\begin{matrix}{{\cos \mspace{14mu} \theta_{2}} = {\cos \mspace{14mu} 140{^\circ}}} \\{{= {{- {0.7}}66044443\mspace{14mu} \ldots}}\mspace{14mu};}\end{matrix}$ $\begin{matrix}{{\theta_{3} = {\theta_{1} + {240{^\circ}}}};} \\{= {{20{^\circ}} + {240{^\circ}}}} \\{= {260{^\circ}}}\end{matrix}$ $\begin{matrix}{{\cos \mspace{14mu} \theta_{3}} = {\cos \mspace{14mu} 260{^\circ}}} \\{{= {{- {0.1}}73648177\mspace{14mu} \ldots}}\mspace{14mu};{and}}\end{matrix}$ $\begin{matrix}{{\cos \mspace{14mu} {\left( {3\theta_{1}} \right)/4}} = {\cos \mspace{14mu} \theta_{1}\mspace{14mu} \cos \mspace{14mu} \theta_{2}\mspace{14mu} \cos \mspace{14mu} \theta_{3}}} \\{= {\left( {\cos \mspace{14mu} \theta_{1\mspace{14mu}}\cos \mspace{14mu} \theta_{2}} \right)\cos \mspace{14mu} \theta_{3}}} \\{= {\left( {{- {0.7}}1984631\mspace{14mu} \ldots}\mspace{14mu} \right)\left( {{- {0.1}}73648177\mspace{14mu} \ldots}\mspace{14mu} \right)}} \\{{= {1/8}};}\end{matrix}$

when a rational number of magnitude 0, −¾, or −3 is to be furthercharacterized, then the fourth, fifth, seventh, eighth, and ninth ofsuch nine equations would apply; whereby those that pertain to the sin θare validated for the particular case when a value of 34.3° becomesassigned to θ₁ as follows:

$\begin{matrix}{{\sin \mspace{14mu} \theta_{1}} = {\sin \mspace{14mu} 34.3{^\circ}}} \\{{= {0.563526048\mspace{14mu} \ldots}}\mspace{14mu};}\end{matrix}$ $\begin{matrix}{\theta_{2} = {\theta_{1} + {120{^\circ}}}} \\{= {{34.3{^\circ}} + {120{^\circ}}}} \\{{= {154.3{^\circ}}};}\end{matrix}$ $\begin{matrix}{{\sin \mspace{14mu} \theta_{2}} = {\sin \mspace{14mu} 154.3{^\circ}}} \\{{= {0.433659084\mspace{14mu} \ldots}}\mspace{14mu};}\end{matrix}$ $\begin{matrix}{{\theta_{3} = {\theta_{1} + {240{^\circ}}}};} \\{= {{34.3{^\circ}} + {240{^\circ}}}} \\{{= {274.3{^\circ}}};{and}}\end{matrix}$ $\begin{matrix}{{\sin \mspace{14mu} \theta_{3}} = {\sin \mspace{14mu} 274.3{^\circ}}} \\{{= {{- {0.9}}97185133\mspace{14mu} \ldots}}\mspace{14mu};}\end{matrix}$ $\begin{matrix}{0 = {{\sin \mspace{14mu} \theta_{1}} + {\sin \mspace{14mu} \theta_{2}} + {\sin \mspace{14mu} \theta_{3}}}} \\{= {{0.563526048\mspace{14mu} \ldots} + {0.4336590\mspace{14mu} \ldots} + {\sin \mspace{14mu} \theta_{3}}}} \\{= {{0.997185133\mspace{14mu} \ldots} + {\sin \mspace{14mu} \theta_{3}}}} \\{\left. {= 0.997185133} \right) - {0.997185133\mspace{14mu} \ldots}} \\{{= 0};}\end{matrix}$ ${\begin{matrix}{{{- 3}/4} = {{\sin \mspace{14mu} \theta_{1}\sin \mspace{14mu} \theta_{2}} + {\sin \mspace{14mu} \theta_{1}\sin \mspace{14mu} \theta_{3}} + {\sin \mspace{14mu} \theta_{2}\sin \mspace{14mu} \theta_{3}}}} \\{= {{\sin \mspace{14mu} \theta_{1}\sin \mspace{14mu} \theta_{2}} + {\left( {\sin \mspace{14mu} \theta_{1}\sin \mspace{14mu} \theta_{2}} \right)\sin \mspace{14mu} \theta_{3}}}} \\{= {{\sin \mspace{14mu} \theta_{1}\sin \mspace{14mu} \theta_{2}} + {\left( {{- \sin}\mspace{14mu} \theta_{3}} \right)\mspace{14mu} \sin \mspace{14mu} \theta_{3}}}} \\{= {{\sin \mspace{14mu} \theta_{1}\sin \mspace{14mu} \theta_{2}} - {\sin^{2}\mspace{14mu} \theta_{3}}}} \\{= {{0.24437819\mspace{14mu} \ldots} - \left( {{- {0.9}}97185133\mspace{14mu} \ldots}\mspace{14mu} \right)^{2}}} \\{= {{0.24437819\mspace{14mu} \ldots} - {(0.99437819\;)\mspace{14mu} \ldots}}} \\{= {{0.24437819\mspace{14mu} \ldots} - \left( {{{0.2}4437819} + {3/4}} \right)}} \\{{= {{- 3}/4}}\;;\mspace{11mu} {{and}\mspace{14mu} {as}\mspace{14mu} a\mspace{14mu} {check}}}\end{matrix} - {\sin \mspace{14mu} {\left( {3\theta_{1}} \right)/4}}} = {\sin \mspace{14mu} \theta_{1}\mspace{14mu} \sin \mspace{14mu} \theta_{2}\mspace{14mu} \sin \mspace{14mu} \theta_{3}}$$\begin{matrix}{{\sin \mspace{14mu} \left( {3\theta_{1}} \right)} = {{- 4}\sin \mspace{14mu} \theta_{1\mspace{14mu}}\sin \mspace{14mu} \theta_{2}\mspace{14mu} \sin \mspace{14mu} \theta_{3}}} \\{= {{+ {0.9}}74761194\mspace{14mu} \ldots}}\end{matrix}$ $\begin{matrix}{{3\theta_{1}} = {{arc}\mspace{14mu} \sin \mspace{14mu} \left( {0.974761194\mspace{14mu} \ldots}\mspace{14mu} \right)}} \\{{= {77.1{^\circ}}},{{or}\mspace{14mu} 102.9{^\circ}}}\end{matrix}$ $\begin{matrix}{{\theta_{1} = {77.1{{^\circ}/3}\mspace{14mu} {or}}},{102.9{{^\circ}/3}}} \\{{= {25.7{^\circ}}},{{{or}\mspace{14mu} 34.3{^\circ}};}}\end{matrix}$

and

lastly, an example is afforded for the particular case when a quadraticirrational number is to be further characterized, such that when:

${{\tan \mspace{14mu} \left( {3\theta_{1}} \right)} = {- \sqrt{3}}};$${{3\theta_{1}} = {\arctan \left( {- \sqrt{3}} \right)}};$3θ₁ = −60^(∘) θ₁ = −20^(∘) $\begin{matrix}{{\tan \mspace{14mu} \theta_{1}} = {\tan \mspace{14mu} \left( {{- 20}{^\circ}} \right)}} \\{{= {{- {0.3}}63970234\mspace{14mu} \ldots}}\mspace{14mu};}\end{matrix}$ $\begin{matrix}{\theta_{2} = {\theta_{1} + {120{^\circ}}}} \\{= {{{- 20}{^\circ}} + {120{^\circ}}}} \\{{= {100{^\circ}}};}\end{matrix}$ $\begin{matrix}{{\tan \mspace{14mu} \theta_{2}} = {\tan \mspace{14mu} 100{^\circ}}} \\{{= {{- {5.6}}7128182\mspace{14mu} \ldots}}\mspace{14mu};}\end{matrix}$ $\begin{matrix}{{\theta_{3} = {\theta_{1} + {240{^\circ}}}};} \\{= {{{- 20}{^\circ}} + {240{^\circ}}}} \\{= {220{^\circ}}}\end{matrix}$ $\begin{matrix}{{\tan \mspace{14mu} \theta_{3}} = {\tan \mspace{14mu} 220{^\circ}}} \\{{= {0.839099631\mspace{14mu} \ldots}}\mspace{14mu};{and}}\end{matrix}$ $\begin{matrix}{{{- \tan}\mspace{14mu} \left( {3\; \theta_{1}} \right)} = {\tan \mspace{14mu} \theta_{1}\mspace{14mu} \tan \mspace{14mu} \theta_{2}\mspace{14mu} \tan \mspace{14mu} {\theta_{3}.}}} \\{= {\left( {\tan \mspace{14mu} \theta_{1}\mspace{14mu} \tan \mspace{14mu} \theta_{2}} \right)\mspace{14mu} \tan \mspace{14mu} \theta_{3}}} \\{= {\left( {2.0641777\mspace{14mu} \ldots}\mspace{14mu} \right)\mspace{14mu} \left( {0.8390996\mspace{14mu} \ldots}\mspace{14mu} \right)}} \\{= {1.732050808\mspace{14mu} \ldots}} \\{{= \sqrt{3}};}\end{matrix}$

and

conversely, whenever trigonometric values of triads θ₁, θ₂, and θ₃become afforded as given quantities, geometric construction patterns canbe approximated which are analogous to the above equations. For example,a unit circle can be drawn which exhibits three radii that emanate fromits origin describing angles of θ°, (θ+120°), and (θ+240°) with respectto its x-axis and terminate upon its circumference. Accordingly, fromthe equation below, the sum of their three ordinate values always mustbe equal to zero, verified algebraically as follows:

$\begin{matrix}{0 = {{\sin \mspace{14mu} \theta_{1}} + {\sin \mspace{14mu} \theta_{2}} + {\sin \mspace{14mu} \theta_{3}}}} \\{= {{\sin \mspace{14mu} \theta_{1}} + {\sin \mspace{14mu} \left( {\theta_{1} + {120}} \right)} + {\sin \mspace{14mu} \left( {\theta_{1} + {240}} \right)}}} \\{= {{\sin \mspace{14mu} \theta_{1}} + \left( {{\sin \mspace{14mu} \theta_{1}\mspace{14mu} \cos \; 120} + {\cos \mspace{14mu} \theta_{1}\sin \; 120}} \right) +}} \\{\left( {{\sin \mspace{14mu} \theta_{1}\mspace{14mu} \cos \mspace{14mu} 240} + {\cos \mspace{14mu} \theta_{1}\mspace{14mu} \sin \mspace{14mu} 240}} \right)} \\{= {{\sin \mspace{14mu} \theta} + {\sin \mspace{14mu} {\theta \left( {{- 1}/2} \right)}} + {\cos \; {\theta \left( {\sqrt{3}/2} \right)}} +}} \\{{{{\sin \mspace{14mu} {\theta \left( {{- 1}/2} \right)}} + {\cos \; {\theta \left( {\sqrt{3}/2} \right)}}};{{such}\mspace{14mu} {that}}}}\end{matrix}$ $\begin{matrix}{0 = {{\sin \mspace{14mu} {\theta \left( {1 - {1/2} - {1/2}} \right)}} + {\cos \mspace{14mu} {\theta \left( {{\sqrt{3}/2} - {\sqrt{3}/2}} \right)}}}} \\{= {{\sin \mspace{14mu} {\theta (0)}} + {\cos \mspace{14mu} {\theta (0)}}}} \\{= 0.}\end{matrix}$

Before even trying to solve the classical problem of the trisection ofan angle, either the designated magnitude of an angle which is intendedto be trisected or some geometric construction pattern which fullydescribes it first needs to be furnished!

To the contrary, if such information instead were to be withheld, thenthe exact size of an angle which is intended to be trisected would notbe known; thereby making it virtually impossible to fulfill the task ofdividing into three equal parts.

In effect, such provision of an a priori condition performs the veryimportant role of identifying exactly which classical problem of thetrisection of an angle is to be solved out of a virtually infinitenumber of possible forms it otherwise could assume depending upon whichdesignated magnitude comes under scrutiny!

For example, attempting to trisect a sixty degree angle solely byconventional Euclidean means poses an entirely different problem thantrying to trisect a seventeen degree angle by means of applying the verysame process.

From an entirely different point of view, whenever a motion relatedsolution for the problem of the trisection of an angle becomesportrayed, it signifies that an actual event has taken place. Such isthe case because some period of time must elapse in order to repositiona trisecting emulation mechanism to a designated setting.

If this were not the case, specifically meaning that an element of timewould not be needed in order to effect trisection, then a motion relatedsolution for the problem of the trisection of an angle thereby could notoccur; simply because without time, there can be no motion!

In support of such straightforward line of reasoning, however, itsurprisingly turns out that a trisecting emulation mechanism furthermorecan portray a stationary solution for the problem of the trisection ofan angle, as well; not as an event, but by sheer coincidence; meaningthat such portrayed solution materializes before time can expire!

The only way this could occur is by having such solution be portrayedbefore an a priori condition becomes specified; thereby suggesting thatsuch solution becomes posed even before defining the full extent problemwhich it already has solved.

Essentially, such stationary solution for the problem of the trisectionof an angle consists of a condition in which the designated magnitude ofan angle which is intended to be trisected just so happens to match theparticular reading that a trisecting emulation mechanism turns out to beprematurely set to before such activity even commences.

The only problem with such stationary solution scenario is that itsprobability of occurrence approaches zero; thereby negating itspractical application. Such determination is computed as such singularreading selection divided by the number all possible readings which suchdevice could be set to, generally comprised of a virtually unlimitednumber of distinct possibilities, and thereby amounting to a ratio whichequates to 1/∞→0.

The input box appearing in such FIG. 2 flowchart, entitled PROBABILISTICPROOF OF MATHEMATIC LIMITATION 10, refers to the specific results whichcan be obtained by realizing that a given angle within a geometricconstruction pattern furthermore must serve the dual role of also beinga trisector for any rendered angle therein whose magnitude amounts toexactly three times its size; thereby signifying that a trisection eventsuccessfully has been performed solely by conventional Euclidean means!

Unfortunately although posing a legitimate solution for the classicalproblem of the trisection of an angle, such rather elementary approachalso proves to be entirely impractical; simply because there is no wayof assuring that such generated rendered angle matches the designatedmagnitude of an angle which is intended to be trisected; as had to bespecified as an a priori condition even before attempting to generatesuch solution!

Since such a priori condition might have specified any of an infinitenumber of possible designated magnitudes, the probability of suchgeometric construction activity proving successful approaches zero, asagain calculated by the ratio 1/∞→0.

Therefore, the practicality of actually attempting to solve suchclassical problem of the trisection of an angle solely by conventionalEuclidean means now easily can be evaluated; whereby any singulargeometric construction pattern which could be generated in such mannerthat the magnitude of its rendered angle amounts to exactly three timesthe size of a given angle, as well as turns out to be equal to adesignated magnitude which previously was identified, because it bears aprobability that approaches zero percent of posing a legitimate solutionfor such classical trisection problem, pretty much should be consideredto be an impossible avenue for obtaining such solution!

Another interpretation is that an angle could be divided into threeequal parts by means of applying only a straightedge and compass to it,but only under the highly unusual condition that an unlimited number ofopportunities become extended, thereby assuring success. Unfortunately,such alternate approach also should be viewed to be quite unacceptablebecause it would take forever to complete.

To follow through with such discussion, it should be mentioned, however,that an approach to solve such classical problem of the trisection of anangle in this very manner already was discovered. As copyrighted inchapter six of my never before published 1976 treatise entitled,Trisection, an Exact Solution, as filed under copyright registrationnumber TXu 636-519, such infinite point solution can trisect in aprecise manner by means of performing a multitude of consecutive angularbisections, all geometrically constructed upon just a single piece ofpaper. Since such solution was authored more than forty years ago, it isincluded herein for purposes of being shared with the general public forthe very first time, but only after formally introducing the fourembodiments of such newly proposed articulating trisection inventionfirst.

In FIG. 2, notice that all five of such parallelogram shaped listedinput boxes, along with the non-iterative, or YES output portion of suchDEFICIENCY MITIGATED 5 decision box, all funnel into a downstreamprocess box which is entitled GEOMETRIC FORMING PROCESS DEVISED 11.

Within such flowchart, although such process box is limited basically totrisection matters, a geometric forming process nevertheless isindicative of a whole gamut of improved drawing pretexts, besides thatEuclidean formulations, which could be developed in order to chartcertain other distinct motions which lie outside of its presentlydiscussed purview, or very narrow scope which hereinafter is to beaddressed in this presentation. Accordingly, it is important to notethat such overall process, at some future date, furthermore could proveto be the source of countless other discoveries which would requireeither a motion related geometric substantiation, and/or an analogoushigher order algebraic solution; thereby evidencing the enormity of ageometric forming capability with regard to its profound influence uponother forms of mathematics.

In 1893, Thomas Alva Edison at long last showcased his kinetoscope.Obviously, such discovery spurred on the development of a cinematicprojector by the Lumiere brothers shortly afterwards. Unfortunately,many instances can be cited in human history in which follow-upinventions of far larger practical importance succeed earlier landmarkcases. Ironically, such type of mishap befell Edison on anotheroccasion, as well; being when he developed a direct current capabilitywhich thereafter became improved upon by Tesla during such time that heintroduced alternating current. Accordingly, one fitting way to suitablyaddress such above described disparity would be to unequivocally statethat due to a series of ongoing technical developments, an entire motionpicture industry eventually became ushered in; whereby a great fanfarefinally arose, as caused by a rather unsuspecting audience who becamemore and more accustomed to witnessing the actual footages of worldevents at the cinema, as opposed to just reading about them in thenewspapers. Over time, the general public began to welcome viewing newsin a more fashionable setting. In retrospect, Kempe's attempts todisclose how to articulate an anti-parallelogram linkage assembly forthe express purpose of performing trisection most certainly appeared toreceive far less critical attention. Whether or not there existed alarge interest in such subject matter is hard to fathom, for justconsider: A full fifteen years prior to Lumiere's actual cinematicprojector debut, dating back all the way to the late 1880's, itobviously would have been very difficult, if not impossible, to revealin sufficient detail to any awaiting crowd, and that much less to onethat might have been gathered some distance way, just how to articulatean anti-parallelogram linkage device in order to satisfactorily performtrisection. Moreover, consider: Had a presentation to this effectsuccessfully been pulled off at that very time period, it more fittinglymight have been mistaken for some sort of magic act! Be that as it may,had there also been a considerable demand levied beforehand, for exampleby some predisposed mathematics party who might have expressed aninterest in viewing such purported trisection capability, it evidentlywould have had very little effect in the overall scheme of things. As itwere, way back in the 1880's, with such industrial community seriouslylagging behind in development, as least in comparison to what actuallyhad become accomplished just ten to fifteen years later, fewer newsorganizations would have been available to disseminate importanttechnical information of that kind. In sharp contrast, only ratherrecently has it truly become possible to pictorially describe just how aKempe anti-parallelogram trisection device actually functions. Intoday's technology, a presentation very easily could be made, merely bymeans of simulating the relative movement of such Kempeanti-parallelogram device within a modem day computer. However, withoutbeing predisposed to such type of information, or even to a lesserextent, thoroughly apprised of such professed trisection capabilities,it most certainly would be very difficult, indeed, to foresee that theoverall technique used to create the very illusion of motion all thoseprior years, merely by means of animating some ragtag assortment ofpictures, or possibly even some collection of photographs whose overallshapes would have been known to differ imperceptibly from one to thenext, furthermore could have been applied to replicate an observedmotion by means of instead animating an entire family of relatedgeometric construction patterns! Hover, had such association truly beenmade those many years before, it well might have contributed tosubstantiating that some articulating prior art mechanism actually couldperform trisection effectively throughout its wide range of devicesettings.

Another possible reason for such noticeable omission could be areticence, or complacency stemming from the fact that, not only hadconventional Euclidean practice proved entirely satisfactory for use onmost prior occasions, but moreover that, up until now, generating asingular drawing pattern was the preferred way to pictorially displayvarious aspects of mathematics.

Unfortunately, as it just so happens to turn out, one of the very fewinstances in which a singular conventional Euclidean practice approachshould not be applied, just for the very reasons expressed above, iswhen attempting to provide the solution for the classical problem of thetrisection of an angle!

As such, it might well be that a recommendation never before was raised,thereby proposing to extend conventional Euclidean practice into ageometric forming process that is fully capable of describing certainmotions, simply because such aforementioned complacency very well by nowactually might have escalated into a full blown reluctance on the partof a seemingly silent majority of mathematical authoritarians toovercome the crippling Euclidean limitation of not being able tobacktrack upon irreversible geometric construction patterns!

With regard to the particular damage levied upon trisection matters overthe years by not otherwise adopting a formal geometric forming process,consider the very first English language trisection involvement, tracingall the way back to a particular drawing which appears on page 309 ofsuch 1897 The Works of Archimedes. Inasmuch as such drawing isaccompanied by a complete accounting of such previously referred toArchimedes proposition, as well as a suitable algebraic proof needed tosubstantiate it, the apparent problem is that such drawing only is asingular geometric construction pattern, thereby applying only to thespecific chord length which appears within its depicted circle. In orderfor such drawing depiction to be fully consistent with such Archimedesproposition and supporting algebraic proof, it should be represented byan entire Euclidean formulation, replete with an infinite number ofother chord lengths which furthermore could be described within suchcircle, and which such Archimedes proposition and supporting algebraicproof also apply to. Without such incorporation, such drawing remainsquite adequate for substantiating the arbitrarily selected chord patternwhich is illustrated therein, but nonetheless remains grosslyimpractical because it cannot represent such infinite number of otherchord shapes and attendant sizes with its circle, and thereby alsoremain subject to the very requirements posed by such includedproposition. Whereas such drawing evidently was presented as aconvention of the time, it must be presumed that it was provided merelyas an example of all of the other possible geometric constructionpatterns which also could have been drawn while still satisfying all ofthe requirements of such proposition. Unfortunately, the key elementthat never was stated therein is that all of such other possiblegeometric construction patterns furthermore must stem from the very samesequence of Euclidean operations that governs such singular drawing, asis represented therein.

Based upon such prior trisection rationale discussion, it becomesapparent that a singular geometric construction pattern can depict onlyone event which takes place during an entire articulation process,thereby representing only a momentary viewing which neither can providean indication of where a particular motion might have originated from,nor where it might have ended up.

Accordingly, such singular drawing format remains somewhat deficientfrom the standpoint that it cannot even define all of the variousgeometries needed to characterize an entire articulated motion!

As such, a singular geometric construction pattern can be likened to astill photograph. Whereas the latter gave birth to the motion pictureindustry, it seems only appropriate that the former should serve as thebasis for an improved geometric approach that becomes capable ofcharacterizing motion!

Such newly proposed geometric forming process capitalizes upon the novelprospect that it requires an entire family of geometric constructionpatterns to adequately represent all of the unique shapes needed torepresent a complete articulation event.

Accordingly, Euclidean formulations can be of service in motion relatedproblems which cannot be fully interpreted by a singular geometricconstruction pattern.

With particular regard to trisection matters, the magnitude of at leastone rendered angle exhibited within any constituent geometricconstruction pattern that belongs to a substantiating Euclideanformulation, quite obviously would need to amount to exactly three timesthe size of its given angle.

Hence, by means of verifying that its outline matches the overall shapeof a corresponding regenerated static image that becomes automaticallyportrayed once a trisecting emulation mechanism becomes properly set,its smaller static image portion thereby could be substantiated toqualify as an associated trisector for such device setting.

As such, a Euclidean formulation, recognizable by its double arrownotation, could dramatically simplify the overall process needed tosubstantiate that some proposed invention has been designed so that itcan perform trisection accurately over a wide range of device settingsand, in so doing, thereby become referred to as a bona fide trisectingemulation mechanism; as duly is depicted in the lower right hand portionof such FIG. 2 flowchart.

Hence, applying such novel geometric forming process in this respectthereby validates that overlapment points, normally considered to bedetrimental because they remain inconspicuous, can be supplanted withintersection points that become fully distinguishable as regeneratedstatic images become automatically portrayed by means of properlysetting trisecting emulating mechanisms

In closing, it should be mentioned that when imposing a controlledmotion, it becomes possible to discern overlapment points; whereby suchEuclidean limitation of otherwise not being able to distinguish them bymeans of backtracking exclusively from a rendered angle within anirreversible geometric construction becomes rectified!

Recapping, an overall explanation just has been afforded for the veryfirst time which maintains that a discernment of overlapment pointsleads to trisection. Hence, it couldn't possibly have been referred toin any prior art.

Moreover, since such explanation alone accounts for how a motion relatedsolution for the problem of the trisection of an angle can be portrayed,prior art couldn't possibly have rendered a differing substantiationthat actually accounts for such professed capabilities.

Any further discussion concerning specific amounts of time which may beneeded to arrange trisecting emulation mechanisms to particular devicesettings are omitted herein because such input is irrelevant whenattempting to substantiate a motion related solution for the problem ofthe trisection of an angle; especially when considering that such timesobviously would vary depending upon a user's dexterity, as well as thevarying distances encountered when going from where such device might betemporarily positioned to a particular device setting.

In conclusion, if the logic proposed in such FIG. 3 Trisection MysteryIteration Processes Table turns out to be entirely true, meaning that aninability to solve the classical problem of the trisection of an angleresults because it impossible to backtrack upon any irreversiblegeometric construction pattern, then it would be utterly senseless toattempt drawing any type of singular geometric construction patternwhatsoever, solely by conventional Euclidean means, in an effort toachieve such ends!

Moreover, when considering that it is necessary to exert a motion inorder to properly set any trisecting emulation mechanism, such warrantedflexure could not, in any way, be fully described solely by a singulargeometric construction pattern!

The process box entitled CLASSICAL PROBLEM OF THE TRISECTION OF AN ANGLESOLUTION DISCREDITED 12 is to serve as the principal focal point withinsuch flowchart, as represented in FIG. 2, where two distinct,independent Euclidean trisection approaches are to be discredited.Below, it should become rather obvious that such second listed approachis an entire reversal of the first:

not only is it impossible to fully backtrack upon any rendered anglewithin a geometric construction pattern whose magnitude amounts toexactly three times the size of its given angle, thereby explaining whythe classical problem of the trisection of an angle cannot be solved;but

conversely, the probability that the magnitude of a rendered anglematches the designated magnitude of an angle that is intended to betrisected approaches zero whenever such rendered angle becomesgeometrically constructed such that its magnitude amounts to exactlythree times the arbitrarily selected size of a given angle.

The fact that a duration of time is needed in order to effect a motionrelated solution for the problem of the trisection of an angleeliminates the possibility that such form of solution potentially mightdouble as a solution for the classical problem of the trisection of anangle. This is because any geometric construction pattern, once drawn,cannot be modified just by applying time to it; thereby affording aprobability that still approaches zero that its overall outline justmight happen to superimpose upon that which otherwise could beautomatically portrayed whenever a static image becomes regenerated bymeans of properly setting any trisecting emulation mechanism.

Moreover, when investigating whether a geometric solution furthermoremight qualify as a solution for the classical problem of the trisectionof an angle, it should be remembered that if extraneous information wereto become introduced into such problem that turns out to be relevant todetermining its solution, then only a solution for some corruptedversion of the classical problem of the trisection of an angle could beobtained; thereby solving an entirely different problem and, in sodoing, discrediting any potential claims that might incorrectly allegethat the classical problem of the trisection of an angle has beensolved.

Lastly, for those remaining skeptics who otherwise would prefer tobelieve that a solution for the classical problem of the trisection ofan angle might yet be specified, all they need to do is disprove that anavailability of overlapment points actually prevents backtracking upon arendered angle within any geometric construction pattern all the wayback to a given angle whose magnitude amounts to exactly one-third ofits size!

In other words, to dispute the new theory that is presented herein, itis now up to them to identify some as yet unidentified geometricconstruction pattern which would enable an angle of virtually anydesignated magnitude they might decide upon to be trisected; whenneither violating the rules which pertain to conventional Euclideanpractice, not introducing any extraneous information which could beconsidered to be relevant to its solution!

Over time, as such ascribed overlapment attribution finally becomesacknowledged to be the real cause for being unable to solve theclassical problem of the trisection of an angle, ongoing analysisthereby could be performed in order to confirm, beyond any shadow ofdoubt, that trisection of an angle of any magnitude cannot be performedsolely by means of applying only a straightedge and compass to it!

The process box entitled SINGULAR DRAWING SOLUTION DISPELLED 13 isincluded in such FIG. 2 flowchart to address the fact that although asingular drawing solution can be described for any regenerated staticimage that automatically becomes portrayed once a trisecting emulationmechanism becomes properly set, designing a device of that type whichhas only one discrete setting would be entirely impractical!

Conversely, any proposed articulating trisection invention that onlyspecifies a singular motion related solution for the trisection of anglecouldn't possibly substantiate a trisection capability for its remainingwide range of settings!

The process box described as SUPPLEMENTAL DEVICE CAPABILITIES SPECIFIED14 is the principal location in such FIG. 2 flowchart where informationpertaining to such MATHEMATICS DEMARCATION 8 input box contributes to anunderstanding that trisecting emulation mechanisms additionally have theaffinity to portray exact lengths that only could be approximated whenperforming geometric construction upon a given length of unity.

Such fact is duly reflected in such FIG. 11 Mathematics DemarcationChart wherein cubic irrational real number types appear only in itsthird column, as headed by the geometric forming process cell; therebyindicating that deliberate motions must be imparted in order portraythem. They can appear either as the ratios of portrayed lengths withrespect to a given length of unity, or as trigonometric propertiesinherent within trisecting angles which become portrayed during certaintrisection events.

For the particular case of the fourth embodiment of such newly proposedinvention, a supplemental device leveling capability also is to bethoroughly described.

Within a right triangle, if the ratio between the length of one of itssides to that of its hypotenuse is cubic irrational, so must be theother. In other words, if one trigonometric property of a right triangleis cubic irrational, so must be all of its trigonometric properties!

It then logically would follow that for any right triangle that exhibitscubic irrational trigonometric properties whose hypotenuse amounts toone unit in length, the lengths of its constituent sides each would haveto be of a cubic irrational value.

Such association enables the lengths of the sides of such right triangleto compensate for each other. With regard to the Pythagorean Theorem,this means that only the sum of the squares of two cubic irrationalvalues can equal a value of one; thereby avoiding the common pitfall ofotherwise attempting to equate such rational unitary value to the squareof a cubic irrational value added to the square of either a rational orquadratic irrational value!

The reason that a right triangle which exhibits cubic irrationaltrigonometric properties truly can be geometrically constructed isbecause of the large number of geometric construction patterns whichexist, all meeting such criteria; whereby the probability of drawingjust one of them out of sheer coincidence increases dramatically.

Attempting to reproduce any one of them just be conventional Euclideanmeans, however, nevertheless would prove fruitless, resulting only in amere approximation thereof; one which might prove suitable when beingconsidered as a duplicate rendering, but not when taking into accountdifferences between them which possibly only would become discernablewell beyond what the capabilities of the human eye could detect.

By finally acknowledging that angles which exhibit cubic irrationaltrigonometric properties actually can be portrayed, their exactmeasurements would become revealed for the very first time, despite thefact that their real values can be described only by decimal patternsthat are never-ending. Perhaps such new found capability very well mightbecome perceived as an uncharted gateway that unfortunately wasoverlooked time and time again in the past!

FIG. 1B presents a very good example of such capability to portrayangles which exhibit cubic irrational trigonometric properties. Therein,angle QPS amounts to exactly sixty degrees. Such sixty degree angle QPSwas chosen because, as stated earlier, its magnitude can be representedexactly by any of the included angles within an equilateral triangle,and thereby can be drawn solely by conventional Euclidean means.

Its associated trisector ∠NMP=∠QMP=∠RMP must be equal to exactlyone-third of its size, amounting to a value which computes to 60°/3=20°.

Upon interpreting FIG. 1B to be representative of a famous as a markedruler arrangement, angle NMP would be twenty degrees in magnitude.

Moreover consider that the notch appearing in its ruler resides awayfrom its endpoint, M, one unit of measurement.

In isosceles triangle NMP, since length MN=length NP=1, it logicallyfollows that twice the cosine of angle NMP would amount to the ratiobetween length MP length MN, whereby the following relationship therebycould be obtained:

${\overset{\_}{MP}/\overset{\_}{MN}} = {2\; {\cos \left( {\angle \; {NMP}} \right)}}$${{\overset{\_}{MP}/1} = {2\; \cos \; 20{^\circ}}};{and}$$\begin{matrix}{\overset{\_}{MP} = {2\left( {{0.9}3969262\mspace{14mu} \ldots}\mspace{14mu} \right)}} \\{= {1.879385242\mspace{14mu} \ldots}}\end{matrix}$

Hence, a cubic irrational value 1.879385242 . . . must be the exactlength of base MP of isosceles triangle NMP; whereby the three dotsnotated after such number indicates that such decimal pattern extends onindefinitely.

Since the cosine of twenty degrees furthermore is a transcendental,number, the above procedure also could distinguish such number types,thereby constituting a subset of cubic irrational numbers.

Once having devised a suitable geometric forming process, it therebybecomes possible to verify that device candidates which wish to qualifyas trisecting emulation mechanisms conform to the various elements whichfunnel into such process box. For example, all devices must be shown tobe fully capable of performing the primary function of regeneratingstatic images, or be bound by the same set of rules. Devices which meetsuch criteria, but thereafter are found to share common design traits,should be categorized as such in order to assure that each itemappearing within any particular group features some fundamentalperformance difference which qualifies it as being individually unique.The TRISECTION INVENTIONS CATEGORIZED 15 process box represents thelocation within such FIG. 2 flowchart where associations of this natureare to be carried out.

The process box therein entitled REQUIREMENTS CHART PREPARED 16 isintended to distinguish that, although CATEGORY I and CATEGORY II priorart devices actually can perform trisection over a wide range of devicesettings, certain aspects of such capability never before werecompletely substantiated. The remainder of such FIG. 2 flowchart,including the decision box entitled DESIGN REQUIREMENTS MET 18, havebeen added for the express purpose of specifying that all of such posedrequirements must be satisfied in order for a proposed design, asdescribed by the process box entitled PROPOSED INVENTION DESIGNREFINEMENT 17, to fully qualify as a trisection emulation mechanism, asitemized in the TRISECTING EMULATION MECHANISM SUBSTANTIATED 19 processbox described therein.

In closing, a novel geometric forming process just has been proposedwhich suitably explains how to rectify a major Euclidean limitation,essentially consisting of an incapability to distinguish overlapmentpoints; as achieved simply by means of imposing a controlled motionwhich makes it possible to discern them!

Although trisection today can be performed because of such identifiedmotion related compensation, were such deleterious behavior otherwise toremain unchecked, then trisection, as sought after by countless futileattempts to solve the famous classical problem of the trisection of anangle still would remain a very illusive problem!

Accordingly it is concluded that a geometric forming process therebyeclipses a rather limited conventional Euclidean practice that has beenin vogue for millennia!

Now that new definitions have been provided, and a resultingcomprehensive methodology, as presented in FIG. 2 has been suitablydescribed, it is due time to account for exactly how a trisectingemulation mechanism operates.

FIG. 12 has been prepared just for this purpose. Such flowchartcommences by means of supplying details to an input box, as entitledDESIGNATED ANGLE SPECIFIED 120 therein. Such specific activity consistsmerely of selecting the designated magnitude of an angle that isintended to become trisected.

The decision box entitled DEVICE NEEDS TO BE SPECIFICALLY ARRANGED 121is where it is to be determined which particular embodiment is to beutilized to perform such anticipated trisection; whereby:

if either such first, second, or fourth embodiment were to be chosen,then the YES route would apply, thereby leading to a process boxentitled DEVICE IS SPECIFICALLY ARRANGED 122 which is where such deviceis to be specifically arranged in accordance with applicable provisions;or

if such third embodiment were to be chosen, then the NO route wouldapply, thereby leading to a process box entitled, DEVICE IS SET 123.

At this stage in the flowchart, such chosen device now should beproperly set to a magnitude which matches the designated magnitude whichfirst was specified.

The next process box entitled, STATIC IMAGE BECOMES REGENERATED 124refers to the fact that by having properly set such device, a specificstatic image became regenerated, a particular portion of which assumedthe overall outline of an actual trisector for such device setting;thereby automatically portraying a motion related solution for theproblem of the trisection of an angle.

Activities which appear inside of the large square shaped dotted lineare those which are to be performed exclusively by any trisectingemulation mechanism which might be placed into use, thereby beingconsidered as properties that are intrinsic to it.

Outside of such trisecting emulation mechanism dotted box, the processbox entitled, TRISECTOR AUTOMATICALLY PORTRAYED 125 is where such motionrelated solution for the trisection of an angle thereafter can bewitnessed.

In connection with such input box entitled MATHEMATICS DEMARCATION 8, asposed in FIG. 2 herein, it previously was mentioned that a Euclideanformulation, each of whose constituent geometric construction patternsexhibits a rendered angle whose magnitude amounts to exactly three timesthe size of its given angle, is to become obtained by means of havingthe value of the sine of any of such rendered angles described by alength of 3 sin θ−4 sin³ θ; thereby conforming to a famous functionexpressed as 3 sin θ−4 sin³ θ=sin (3θ).

With regard to the very limited scope of trisection covered in thispresentation, it should suffice to say that discussions below are tobegin by significantly pointing out that the pretext of a Euclideanformulation just so happens to be conducive to physically describingvarious equations which have an infinite number of solutions!

Perhaps the most relevant of these, as specified below, assume the formof three very famous cubic expressions which address trisection by meansof relating trigonometric properties of one angle of variable size toanother whose magnitude always amounts to exactly three times its size:

cos   (3θ) = 4cos³θ − 3cos  θ;sin   (3 θ) = 3 sin  θ − 4 sin³ θ; and${\tan \mspace{14mu} \left( {3\; \theta} \right)} = {\frac{{3\tan \mspace{14mu} \theta} - {\tan^{3}\theta}}{1 - {3\; \tan^{2}\theta}}.}$

Whenever the magnitude of an angle that is algebraically denoted to beof size 3θ becomes supplied as a given quantity in any of such threecubic expressions, then such algebraic relationship truly would typifytrisection!

This is because, a corresponding magnitude of θ, being an exacttrisector of such given 3θ value, then could be computed simply by meansof dividing such given value by a factor of three; thereby enabling adetermination of the constituent trigonometric properties, as specifiedabove.

For example, for the particular condition when it is given that:

3θ = 75^(∘) $\begin{matrix}{\theta = {75{{^\circ}/3}}} \\{= {25{^\circ}}}\end{matrix}$ cos   θ = 0.906307787 3 cos   θ = 2.7189233614 cos³  θ = 2.977742406; $\begin{matrix}{{\cos \left( {3\theta} \right)} = {{4\cos^{3}\mspace{14mu} \theta} - {3\; \cos \mspace{14mu} \theta}}} \\{= {{2.977742406} - {2.718923361}}} \\{{= 0.258819095}\;;}\end{matrix}$

and

As a check, 3θ=75°√

Conversely, if an infinite number of magnitudes of θ were to becomesupplied as given values instead, each of such three algebraicrelationships thereby could be suitably represented by means ofdeveloping a newly established Euclidean formulation that fully coulddistinguish it!

This is because all three of such above cited cubic expressions arecontinuous and their respective right-hand terms furthermore aregeometrically constructible.

To aptly demonstrate this, a Euclidean formulation, as posed in FIG. 13,has been developed to suitably represent such famous cubic relationshipsin (3θ)=3 sin θ−4 sin³ θ; wherein any geometric construction patternbelonging to thereby would exhibit a discrete value of sin (3θ) for eachand every selected real sin θ value existing within the range of −1 to+1.

The governing sequence of Euclidean operations for such new Euclideanformulation is specified as follows:

given angle VOO′ is geometrically constructed of an arbitrarily selectedmagnitude that algebraically is denoted as θ such that its side OO′exhibits the same length as its side OV;

side OV is designated to be the x-axis;

a y-axis is drawn, hereinafter represented as a straight line whichpasses through vertex O of given angle VOO′ and lies perpendicular tosuch x-axis;

a UNIT CIRCLE ARC becomes geometrically constructed, hereinafter to berepresented as a portion of the circumference of a circle drawn aboutcenter point O whose radius is set equal in length to OV, therebyenabling it to pass through points V and O′, both of which previouslyhave been designated as respective termination points of angle VOO′;

point T thereafter becomes designated as the intersection between suchUNIT CIRCLE ARC and such geometrically constructed y-axis;

a straight line which passes through point O is drawn at forty-fivedegree angle counterclockwise to such x-axis;

another straight line which passes through point O is drawn making athree-to-one slope with the +x-axis;

a horizontal straight line is drawn which passes through point O′ andthereby lies parallel to the x-axis;

the juncture between such horizontal straight line and the y-axisbecomes designated as “sin θ”, thereby denoting its vertical distanceabove such x-axis;

a vertical straight line is drawn so that it remains parallel to they-axis while passing through the intersection made between suchforty-five degree straight line and such horizontal straight line;

the horizontal distance such vertical straight line resides to the rightof such y-axis also thereby is to be designated as “sin θ” along suchx-axis;

a second vertical straight line is drawn which passes through coordinatepoint V, thereby being tangent to such previously drawn UNIT CIRCLE ARC;

a slanted straight line is drawn which originates at point O and passesthrough the intersection point made between such second verticalstraight line and such horizontal straight line;

the angle which such slanted straight line makes with the x-axis becomesdesignated as “ω”, not to be confused with angle VOO′ amounting to aslightly larger magnitude of θ;

a second horizontal straight line is draw which passes through theintersection point made between such slanted straight line and suchvertical straight line;

the juncture of such second horizontal straight line with the y-axisbecomes designated as “h₁”, thereby denoting its unknown verticaldistance above point O;

a second slanted straight line is drawn which extends from point O tothe intersection point made by such second horizontal straight line withsuch second vertical straight line;

the angle which such second slanted straight line makes with the x-axisthereafter becomes designated as “φ”;

a third horizontal straight line is drawn so that it passes through theintersection point made between such second slanted straight line andsuch vertical straight line;

the juncture of such third horizontal straight line with the y-axisbecomes designated as “h₂”, thereby denoting its unknown verticaldistance above point O;

a fourth horizontal straight line is drawn so that it passes through theintersection point made between such straight line which exhibits a 3:1slope with respect to the x-axis and such vertical straight line;

the juncture which such fourth horizontal straight line makes with they-axis becomes denoted as “3 sin θ”, thereby distinguishing its verticaldistance above point O;

a fifth horizontal straight line is drawn at a distance directly belowsuch fourth horizontal straight line which measures four times theheight which such third horizontal straight line resides above suchx-axis, algebraically denoted therein as “4h₂”;

the juncture which is made between such fifth horizontal straight lineand the y-axis becomes designated as “sin (3θ)”, thereby denoting itsvertical distance above point O; and

the intersection point of such fifth horizontal straight line with suchUNIT CIRCLE ARC becomes designated as point U′.

The proof for such FIG. 13 Euclidean formulation is provided below:

$\begin{matrix}{{\tan \mspace{14mu} \omega} = {{{h_{1}/\sin}\mspace{14mu} \theta} = {\sin \mspace{14mu} {\theta/1}}}} \\{{h_{1} = {\sin^{2}\mspace{14mu} \theta}};}\end{matrix}{\quad {{{\begin{matrix}{{\tan \mspace{14mu} \phi} = {{{h_{2}/\sin}\mspace{14mu} \theta} = {h_{1}/1}}} \\{= {\sin^{2}\mspace{14mu} {\theta/1}}}\end{matrix}h_{2}} = {{\sin^{3}\mspace{14mu} \theta 4h_{2}} = {4\; \sin^{3}\mspace{14mu} \theta}}};\begin{matrix}{{\sin \mspace{14mu} \left( {3\theta} \right)} = {{3\; \sin \mspace{14mu} \theta} - {4\; \sin^{3}\mspace{14mu} \theta}}} \\{{= {{3\sin \mspace{14mu} \theta} - {4h_{2}}}};}\end{matrix}}}$

and

since point U′ lies upon such UNIT CIRCLE ARC and exhibits a sin (3θ)ordinate value, radius OU′ must reside at an angle of 3θ with respect tothe x-axis.

Accordingly, FIG. 13 distinguishes an entire family of geometricconstruction patterns, all generated from the very same sequence ofEuclidean operations as stipulated above; with the only exception beingthat the respective magnitudes of given angle VOO′ becomes slightlyaltered each time a new geometric construction pattern becomes drawn.

Based upon a reasoning that such famous cubic relationship sin (3θ)=3sin θ−4 sin³ θ actually can be fully distinguished by an entire familyof geometric construction patterns which together comprise such newlyproposed Euclidean formulation, as posed in FIG. 13, it theoreticallymight become possible to devise yet another rather crude, or cumbersome,trisecting emulation mechanism which, due to a considerable increase inits number of overall working parts, obviously would be considered tolie far beyond the very scope of this presentation. In order to becomefeasible, however, a newly fashioned device of such type would have tobe designed so that when it becomes articulated by means of rotating itsaxis U′ circumferentially about axis O in accordance with such doublearrow notation as expressed in FIG. 13, such motion additionally couldbe replicated by means of animating the conglomeration of geometricconstruction patterns which belong to such Euclidean formulation insuccessive order.

In conclusion, any algebraic determination that can be made by means ofrelating like trigonometric properties that exist between one value andanother that amounts to exactly three times its magnitude, as specifiedin such three cited famous cubic expressions, furthermore can be fullydescribed by a geometric construction pattern which belongs to one ofthree Euclidean formulations which could be developed to characterizethem.

For example, if a particular value of 1.119769515 radians were to beaccorded to θ, then an algebraic determination could be made, as followsof 3θ, which furthermore fully could be described by a singulargeometric pattern which belongs to such newly proposed Euclideanformulation, as posed in FIG. 13:

θ = 1.119769515  radians sin   θ = 0.9; and $\begin{matrix}{{\sin \mspace{14mu} \left( {3\theta} \right)} = {{3\sin \mspace{14mu} \theta} - {4\sin^{3}\mspace{14mu} \theta}}} \\{= {{3\left( {0{.9}} \right)} - {4(0.9)^{3}}}} \\{= {2.7 - {4\left( {{0.7}29} \right)}}} \\{= {2.7 - 2.916}} \\{{= {{- 0}{.216}}};}\end{matrix}$ $\begin{matrix}{{3\; \theta} = {\pi + {{0.2}17715891}}} \\{= {3.359308545}} \\{= {3{\left( {{1.1}19769515} \right).}}}\end{matrix}$

Such above furnished overall detailed accounting explains exactly whyall three of such previously cited famous cubic expressions remainincredibly important!

More particularly, this is because each of such three expressions can beconsidered to be a distinctive format type, in itself, one thatfurthermore can be broken down into an infinite number of uniquerelationships that have three cubic roots each.

Such scenario is far different than what transpires with respect todiscontinuous functions, as are about to be discussed in detail next.

Also in connection with such input box entitled MATHEMATICS DEMARCATION8, as posed in FIG. 2, it previously was mentioned that a graph is tobecome developed that distinguishes between the continuity of such wellknown cubic function 4 cos³ θ−3 cos θ=cos (3θ) and the discontinuitythat very clearly accompanies the function (4 cos³ θ−6)/(20 cos θ)=cos(3θ).

FIG. 14 is intended to make clear such distinction.

Its top legend identifies the path charted by a curve for such firstfamous cubic function, algebraically expressed as y=4 cos³ θ−3 cos θ=cos(3θ) wherein:

abscissa values in x signify cos θ magnitudes; and

ordinate values in y signify cos (3θ) magnitudes.

Such well known curve is shown to be continuous within the specificrange of −1≤x≤+1, thereby accounting for all real number values of cosθ.

The second legend therein identifies the particular function y=(4 cos³θ−6)/(20 cos θ) wherein abscissa values in x again signify cos θmagnitudes. Such curve also is shown to be continuous in the same range,except for the fact that it is discontinuous at x=0. Notice that as thevalue of x, or cos θ, nears zero from a negative perspective, thecorresponding value of y approaches positive infinity, and as it nearszero from the positive side, the corresponding value of y approachesnegative infinity; thereby maintaining a one-to-one relationship betweenx and y values all along its overall path.

Where the curves identified by such first and second legends intersect,they can be equated due to the fact that they exhibit both x values ofequal magnitude, as well as y values of equal size. Algebraically thiscan be expressed by the equation y=(4 cos³ θ−6)/(20 cos θ)=cos (3θ), astypified by the third legend, as displayed in FIG. 14.

Hence, such intersection points, shown to be positioned at the centersof such four large circles drawn therein, locate positions where (4 cos³θ−6)/(20 cos θ)=cos (3θ).

By then substituting 4 cos³ θ−3 cos θ for cos (3θ), as shown below, thefollowing fourth order equation can be obtained, along with adetermination of the four associated roots for cos θ and other relevantquantitative details:

${{{4\cos^{3}\theta} - {3\cos \mspace{14mu} \theta}} = \frac{{4\cos^{3}\theta} - 6}{20\; \cos \; \theta}};$

and

via cross multiplication,

(4 cos³θ−3 cos θ)(20 cos θ)=4 cos³θ−6;

80 cos⁴θ−60 cos²0=4 cos³θ−6;

80 cos⁴θ−4 cos³θ−60 cos²θ+6=0; and

cos⁴θ− 1/20 cos³θ−¾ cos²θ+ 3/40=0.

Values of the roots of such quartic equation are provided in FIG. 15.The first column therein, as headed by the term VALUE, contains variousentries of algebraic significance. For each of such five listed entries,corresponding values are cited each of the four the roots θ₁, θ₂, θ₃,and θ₄ which appear as headings in the following four columns. Noticethat for each of such particular values of θ, as specified in the secondline item therein, a respective value of cos (3θ) appears in the fifthline item therein which is equal to the value of (4 cos³ θ−6)/(20 cosθ), as it appears in the sixth line item therein.

In conclusion, the cos (3θ)=(4 cos³ θ−6)/(20 cos θ) quartic functionclearly qualifies as being discontinuous because it consists of onlyfour distinct points, as are identified by circles appearing in such ofFIG. 14.

With particular regard to the two continuous curve representations drawnin FIG. 14, a Euclidean formulation could be generated, whereby each ofthe singular geometric construction patterns which belong to it can bealgebraically determined; three examples of which are presented directlybelow:

at  x = cos   θ = 1; $\begin{matrix}{y = {\left( {{4\; \cos^{3}\mspace{14mu} \theta} - 6} \right)/\left( {20\; \cos \mspace{20mu} \theta} \right)}} \\{= {\left\lbrack {{4(1)} - 6} \right\rbrack/\left\lbrack {20(1)} \right\rbrack}} \\{= {\left( {4 - 6} \right)/20}} \\{= {{- 2}/20}} \\{{= {{- 1}/10}};}\end{matrix}$ at  x = cos   θ = 1/2; $\begin{matrix}{y = {\left( {{4\; \cos^{3}\mspace{14mu} \theta} - 6} \right)/\left( {20\; \cos \mspace{20mu} \theta} \right)}} \\{\left. {= \left\lbrack {{(4)\left( {1/2} \right)^{3}} - 6} \right)} \right\rbrack/\left\lbrack \left( {20{x\left( {1/2} \right)}} \right\rbrack \right.} \\{\left. {= \left\lbrack {{(4)\left( {1/8} \right)} - 6} \right)} \right\rbrack/10} \\{\left. {= \left( {{1/2} - 6} \right)} \right\rbrack/10} \\{\left. \left. {= {- 5.5}} \right) \right\rbrack/10} \\{{= {{- 0}{.55}}};}\end{matrix}$ at  x = cos   θ = −1/2; and $\begin{matrix}{y = {\left( {{4\; \cos^{3}\mspace{14mu} \theta} - 6} \right)/\left( {20\; \cos \mspace{14mu} \theta} \right)}} \\{= \left\lbrack {{(4)\left( {{- 1}/2} \right)^{3}} - {6/\left\lbrack \left( {20{x\left( {{- 1}/2} \right)}} \right\rbrack \right.}} \right.} \\{\left. {= \left\lbrack {{(4)\left( {{- 1}/8} \right)} - 6} \right)} \right\rbrack/{- 10}} \\{{\left. {= \left( {{{- 1}/2} - 6} \right)} \right\rbrack/{- 1}}0} \\{= {{- \left( {6{.5}} \right)}/{- 10}}} \\{= {0.65.}}\end{matrix}$

Naturally any geometric construction pattern which possibly could bedrawn which belongs to such Euclidean formulation would identify just asingle point which lies upon the two curve potions represented by thesecond legend in FIG. 14.

Above, the length (½)³ would be geometrically constructed in much thesame fashion as was the sin³ θ in FIG. 13. The development of suchenvisioned Euclidean formulation would encompass first generating alength which is equal to (½)², solely by conventional Euclidean means;produced in similar manner to length h₁, as it appears therein. Fromsuch length, another length representative of the algebraic expression(½)³ would become drawn, similar to h₂, as it appears therein.

From the above calculations, it should become rather clear that anentire family of geometric construction patterns could be drawn for thefunction y=(4 cos³ θ−6)/(20 cos θ). The corresponding sequence ofEuclidean operations needed to conduct such activity could be obtainedmerely by administering the formula represented on the right hand sideof the equation given above, thereby represented as (4 cos³ θ−6)/(20 cosθ); whereby only the value of cos θ would be altered in during suchdevelopment.

Each respective length of the ordinate value y then could be drawn byway of the proportion y/1=(4 cos³ θ−6)/(20 cos θ), thereby producingsuch length ‘y’ by means of applying only a straightedge and compass.

As such, the function y=(4 cos³ θ−6)/(20 cos θ) could be fully describedby yet another entirely separate Euclidean formulation. Even though eachof such generated geometric construction patterns belonging to suchEuclidean formulation most certainly would not relate trigonometricvalues of angles to those of angles which amount to exactly one-thirdtheir respective size, it nevertheless would be possible to design anentirely new invention whose distinctive flexure, maybe even being aharmonic motion, could be replicated by means of animating the entirefamily of geometric construction patterns which belong to such newlydevised Euclidean formulation in successive order.

Obviously, such types of involvements inevitably should serve asbuilding blocks for mathematics!

More specifically stated, a novel assortment of sundry mechanicaldevices that exhibit capabilities well beyond those of trisectingemulation mechanisms whose fundamental architectures during flexureregenerate static images that automatically portray overall geometriesthat furthermore can be fully described by Euclidean formulationsadditionally can be quantified algebraically!

In this vein, prior claims made in connection with such FIG. 11Mathematics Demarcation Chart, now are to be somewhat bolstered bytheorizing that the very formats expressed by algebraic equations giveclear indication of the types of geometric construction practices theysupport.

Such explanation begins with what clearly is known concerning any linearfunction of the form y=mx+b.

Its geometric construction counterpart consists merely of locating asecond point which lies a magnitude that algebraically is denoted as ‘b’either directly above or below a first point, depending upon the signplaced in front of such coefficient. For example, in the equationy=6x−3, such second point would be situated exactly three units ofmeasurement below such first point. In order to complete such singulargeometric construction pattern, a straight line next would need to bedrawn which passes through such second described point and furthermoreexhibits a slope, ‘m’, whose rise and run values could be depicted asthe sides of a right triangle, the ratios of whose mutual lengths amountto such magnitudes.

Second order functions of a singular variable cannot be fully describedby a geometric construction process, thereby necessitating instead thatthey be fully charted by means of plotting a y value that appears upon aCartesian coordinate system that becomes algebraically determined foreach x value belonging to such function.

However, conventional Euclidean practice can be of assistance indetermining the roots of quadratic functions. For example, consider anentire set of parabolic functions whose overall format type therebycould be expressed as ax²+bx+c=y.

For any specific values which its coefficients might be respectivelyassigned, a singular algebraic function belonging to such format typewould become specified. Its roots would indicate where such singularcurve crosses the x-axis; but only could when the variable ‘y’ withinsuch function amounts to zero; hence becoming representative of aquadratic equation which instead would belong to another simplifiedformat type, algebraically expressed as ax²+bx+c=0 which would typify asubset of such parabolic function format type.

By means of referring back to the previous discussion regarding suchinput box entitled MATHEMATICS DEMARCATION 8, as posed in FIG. 2, notethat it was mentioned that a geometric construction pattern that isrepresentative of the famous Quadratic Formula z_(R)=(−b±√{square rootover (b²−4ac)})/2a would be created to resolve the parabolic equation of−0.2x²+0.4x+0.75=0 belonging to such ax²+bx+c=0 format type.

Herein, FIG. 16 represents such very solution.

The very sequence of Euclidean operations from which such singulargeometric construction pattern is derived is provided directly below:

a square each whose sides is of unit length is drawn;

a right triangle is inscribed within it such that:

its first side begins at one of the corners of such square, extends alength of 0.75, or ¾ of a unit from it, and becomes drawn so that italigns upon a side of such square, thereafter becoming algebraicallydenoted as being of length ‘c’ therein;

its second side, drawn at a right angle away from the endpoint of suchfirst side, is to be of unit length also such that its endpoint residessomewhere along the opposite side of such previously drawn square; and

its hypotenuse then is to become drawn;

a straight line of length of 0.8 units which extends from a point whichresides somewhere upon the first side of such previously drawn righttriangle that is parallel to its second side, and terminates somewherealong its hypotenuse is to be drawn as follows:

a straight line reference becomes drawn that lies parallel the firstside of such previously drawn right triangle and resides 0.8 units inlength above it;

from the intersection point of such straight line reference and thehypotenuse of such previously drawn right triangle, another straightline is drawn that is perpendicular to such straight line reference;

such 0.8 units in length which spans the distance between the first sideof such previously drawn right triangle and such straight line referenceis to be algebraically denoted as ‘−4a’ therein; and

the span of the first side of such previously drawn right triangle whichextends from its beginning point to where it intersects such straightline which was drawn to be of 0.8 units in length thereby can bealgebraically denoted to be of a length ‘−4ac’ due to the fact that itrepresents a corresponding side belonging to another right trianglewhich is similar such previously drawn right triangle, thereby meetingthe proportion c/1=−4ac/−4a;

a semicircle is drawn whose diameter aligns upon the side of such squarethat the first side of such previously drawn right triangle also alignswith whose circumferential portion lies outside of such square;

such 0.8 unit straight line next is to be extended below the side ofsuch square until it meets such previously drawn circumferentialportion, from which two more straight lines are to be drawn, eachterminating at a lower corner of such square, thereby describing asecond right triangle whose hypotenuse then can be denoted as √{squareroot over (−4ac)}, since is squared value is equal to the area of therectangle inscribed in such square whose sides are of unit and −4acrespective lengths by virtue of the Pythagorean Theorem;

the remaining side of such newly drawn right triangle, as appearingwithin such previously drawn semicircle, becomes extended a distancethat amounts to 0.4 units in length such that the circumference of awhole circle can be drawn about its new endpoint, being of a radius thatthereby can be algebraically denoted to be of length ‘b’ therein;

a straight line then is drawn which extends from the beginning of thefirst side of such previously drawn right triangle that terminates atthe center point of such whole circle, thereby being algebraicallydenoted to be of length √{square root over (b²−4ac)} as determined byPythagorean Theorem, once realizing that it represents the hypotenuse ofyet another right triangle whose respective sides are of lengths b and√{square root over (−4ac)};

such newly drawn straight line then becomes extended until it reachesthe far circumference of such circle, thereby to become algebraicallydenoted to be of overall length b+√{square root over (b²−4ac)};

its span extending from the beginning of the first side of suchpreviously drawn right triangle to the near circumference of such circlethereby becomes algebraically denoted to be of length −b+√{square rootover (b²−4ac)};

another straight line then is drawn which passes through the corner ofsuch previously drawn square upon which the vertex of such previouslydrawn right triangle was geometrically constructed, and its first sidebegan, which furthermore lies perpendicular to the diameter of suchnewly drawn circle which is shown, being a total length of unity suchthat 0.4 units of such overall length resides to right side of suchdiameter, thereby becoming algebraically denoted to be of length −2a;

with respect to such last drawn straight line:

a straight line is drawn perpendicular to its left termination point;and

two more straight lines are drawn emanating from its rightmosttermination point, each of which passes through respective locationswhere the diameter drawn for such circle intersects its circumference;

the longer cutoff made upon such lastly drawn perpendicular straightline thereby is algebraically denoted to be of length x₁, signifying anoverall length whose magnitude is equal to the value of the first rootof such given quadratic function −0.2x²+0.4x+0.75=y, as determined bythe respective sides of two right triangles that establish theproportion x₁/1=(b+√{square root over (b²−4ac)})/−2a, thereforeamounting to x₁=(−b−√{square root over (b²−4ac)})/2a; and

the shorter cutoff made upon such lastly drawn perpendicular straightline thereby is algebraically denoted to be of length −x₂, signifying anoverall length whose magnitude is equal to the negative value of thesecond root of such given quadratic, as determined by the respectivesides of two right triangles that establish the resulting proportion−x₂/1=(−b+√{square root over (b²−4ac)})/−2a, thus amounting tox₂=(−b+√{square root over (b²−4ac)})/2a.

Likewise, a cubic functions of a single variable also cannot be fullydescribed by a single geometric construction pattern, but insteadrequires an entire Euclidean formulation to describe what otherwisewould need to become fully plotted by means of algebraically determininga value of y for each x value belonging to such function; as is the casefor the either of the continuous cubic curves which are charted in FIG.14.

Notice that when interpreting such continuous cubic function y=(4 cos³θ−6)/(20 cos θ):

when reading from right to left, it indicates an entire family of uniquegeometric construction patterns, each of which can be generated by meansof applying the very same sequence of Euclidean operations, whereby onlythe magnitude of its given value, cos θ, becomes slightly altered; but

when otherwise going from left to right, it becomes indicative of acertain motion which could be imparted by some mechanical device whosefundamental architecture during flexure can be replicated by means ofanimating a Euclidean formulation which could fully describe itsconstituent overall shapes. That is to say, a geometric forming processwhich should be incorporated into the fold of mathematics cancharacterize trisection for virtually any of the equations containedwithin the three very famous cubic curves expressed above!

As such, a sequel, or follow-on development, being one that presently isconsidered to be well beyond the very limited scope postulated herein,might entail placing parameters of time within continuous algebraiccubic functions, thereby opening up an entirely new gateway formathematical investigation; principally because motion cannot transpirewithout it.

It is in this area of discussion that perhaps the greatest confusionabounds concerning trisection!

In order to suitably avoid its pitfalls, it becomes necessary to poseone last riddle which finally should fully expose any disturbing mythsthat yet might be perpetuated by such great trisection mystery.

The last riddle is: Can the classical problem of the trisection of anangle actually be solved after gaining an understanding of the rolewhich algebraic expressions play in the determination of the magnitudeof a trisector for an angle of virtually any designated magnitude?

Again, such answer, most emphatically, turns out to be a resounding no!

Such above proposed determination can be substantiated by examining theproceedings associated with a cubic equation containing a singlevariable which becomes resolved by means of simultaneously reducing itwith respect to another cubic equation of a single variable whichharbors a common root, whereby such algebraic process enables vitalinformation to be converted into second order form.

Naturally, such algebraic approach cannot solve the classical problem ofthe trisection of an angle!

However, it can serve to justify that there is a certain order withinmathematics that most certainly should be exposed for the benefit ofmankind!

As a relevant example of this, one of the three famous cubic functionscited above is to undergo such simultaneous reduction process, wherein ζis to denote the particular value of the tangent of a designatedmagnitude of an angle, 3θ, that is about to be trisected; therebybecoming algebraically expressed as tan (3θ). Since such famous cubicequations can track trigonometric relationships which exist betweenvarious given angles and those amounting to exactly three times theirrespective sizes, such previously mentioned common root, denoted asz_(R), is to represent corresponding values of tan θ, thereby enablingthe following algebraic cubic equation expressions to be reformatted asfollows:

whereas, tan (3θ)=(3 tan θ−tan³ θ)/(1−3 tan²θ);

then, ζ=(3z _(R) −z _(R) ³)/(1−3z _(R) ²)

ζ(1−3z _(R) ²)=3z _(R) −z _(R) ³

z _(R) ³=3z _(R)−ζ(1−³ z _(R) ²).

In order to perform such simultaneous reduction, a generalized cubicequation format type of the form z³+βz²+γz+δ=0 now is to becomeintroduced, as well.

In order to determine what common root values any of such equationswhich belong to such generalized cubic equation format type share incommon, in such above equation:

z ³ +βz ² +γz+δ=0;

z _(R) ³ +βz _(R) ² +γz _(R)+δ=0; and

z _(R) ³=−(βz _(R) ² +γz _(R)+δ).

Such format type is to be referred to as the generalized cubic equationbecause its accounts for virtually every possible equation that a cubicequation of a single variable could possibly assume!

Since such famous tangent cubic function can be arranged as z_(R)³−3ζz_(R) ²−3z_(R)+ζ=0, it must be a subset of such generalized cubicequation for the specific case when coefficient β=−3ζ, γ=−3, and δ=ζ.

As, I'm sure the reader by now must have guessed, the significance ofsuch association is that both equation formats thereby must bear acommon root!

Moreover, the term format, as addressed above, applies to a whole familyof equations that exhibit identical algebraic structures, but differonly in respect to the particular values of the algebraic coefficientsthey exhibit!

Such mathematical phenomenon occurs because the uncommon roots of eachparticular equation belonging to such generalized cubic equation format,when arranged in certain combinations with common roots, z_(R), whichthey share with respective equations that belong to such famous tangentcubic equation format, actually determine such other coefficient values,as will be more extensively explained below.

By equating z_(R) ³ terms, the following quadratic equationrelationships can be obtained by means of removing mutual cubicparameters:

3z_(R) − ζ(1 − 3z_(R)²) = −(βz_(R)² + γz_(R) + δ) = z_(R)³(3ζ + β)z_(R)² + (3 + γ)z_(R) + (δ − ζ) = 0a  z_(R)² + b  z_(R) + c = 0; and(3ζ + β)z_(R)² + (3 + γ)z_(R) + (δ − ζ) = 0${z_{R}^{2} + {\frac{3 + \gamma}{{3\zeta} + \beta}z_{R}} + \frac{\delta - \zeta}{{3\zeta} + \beta}} = 0$z_(R)² + b^(′)  z_(R) + c^(′) = 0.

Such last alteration, amounting to the division of each containedcoefficient by a factor of ‘a’, gives an indication of how to furthermanipulate algebraic equation results in order to realize theirgeometric solutions in a more efficient manner, leading to anabbreviated Quadratic Formula of the form z_(R)=(−b+√{square root over(b²−4ac)})/2a=[−b′+√{square root over(b′²−4(1)(c′))}]/2(1)(½)(−b′±√{square root over (b′²−4c′)}).

Obviously, such abbreviated Quadratic Formula then applies only toquadratic equations of a singular variable whose squared termcoefficients are equal to unity!

In order to simultaneously reduce two cubic equations in a singlevariable which share a common root, their remaining root values must bedifferent.

To demonstrate how this works, a generalized cubic equation is to bedetermined whose uncommon roots, for the sake of simplicity exhibitvalues of z_(S)=3 and z_(T)=4.

For the example which is about to be presented below, a common rootvalue of z_(R)=√{square root over (5)} is to be assigned because it isof quadratic irrational magnitude, and thereby can be geometricallyconstructed directly from a given length of unity, thereby representingthe length of the hypotenuse of a right triangle whose sides are oflengths 1 and 2, respectively.

As such, the magnitude of (could be determined merely by means ofcomputing the overall value associated with (3z_(R)−z_(R) ³)/(1−3z_(R)²) (3√{square root over (5)}−5√{square root over (5)})/(1−3×5)=√{squareroot over (5)}/7.

Notice that such calculation furthermore must be of quadratic irrationalmagnitude, thereby enabling such length to be represented as the verystarting point within an upcoming geometric construction process.

Accordingly, such famous cubic relationship in a single variable z_(R)³−3ζz_(R) ²−3z_(R)+ζ=0 would assume the particular form z_(R) ³−3(√{square root over (5)}/7) z_(R) ²−3z_(R)+√{square root over (5)}/7=0.

As for such generalized cubic equation, since it can be stated that:

z−z _(R)=0;

z−z _(S)=0; and

z−z _(T)=0.

By thereafter multiplying such three equations together, the followingalgebraic expression could become obtained:

(z−z _(R))(z−z _(S))(z−z _(T))=0; or

z ³−(z _(R) +z _(S) +z _(T))z ²+(z _(R) z _(S) +z _(R) z _(T) +z _(S) z_(T))z−z _(R) z _(S) z _(T)=0; and

z ³ +βz ² +γz+δ=0.

By equating coefficients of like terms, the following threerelationships can be determined:

$\begin{matrix}{\beta = {- \left( {z_{R} + z_{S} + z_{T}} \right)}} \\{= {- \left( {\sqrt{5} + 3 + 4} \right)}} \\{{= {- \left( {\sqrt{5} + 7} \right)}};}\end{matrix}$ $\begin{matrix}{\gamma = {{z_{R}z_{S}} + {z_{R}z_{T}} + {z_{S}z_{T}}}} \\{= {{\left( \sqrt{5} \right)\left( {3 + 4} \right)} + {3(4)}}} \\{{= {{7\sqrt{5}} + 12}};{and}}\end{matrix}$ $\begin{matrix}{\delta = {{- z_{R}}z_{S}z_{T}}} \\{= {{- \left( \sqrt{5} \right)}\left( {3\left( (4) \right.} \right.}} \\{= {{- 12}{\sqrt{5}.}}}\end{matrix}$

Such generalized cubic equation format would be z³−(√{square root over(5)}+7) z²+(7√{square root over (5)}+12) z−12√{square root over (5)}=0.

Accordingly:

$\begin{matrix}{b^{\prime} = \frac{3 + \gamma}{{3\zeta} + \beta}} \\{= \frac{3 + \left( {{12} + {7\sqrt{5}}} \right)}{{3\left( {\sqrt{5}/7} \right)} - \left( {\sqrt{5} + 7} \right)}} \\{= \frac{{15} + {7\sqrt{5}}}{- \left( {{4\; {\sqrt{5}/7}} + 7} \right)}} \\{{= {- \left( \frac{105 + {49\sqrt{5}}}{{4\sqrt{5}} + 49} \right)}};}\end{matrix}$ $\begin{matrix}{b^{\prime^{2}} = \frac{{105^{2}} + {210(49)\sqrt{5}} + {49^{2}(5)}}{{16(5)} + {8(49)\sqrt{5}} + {49^{2}}}} \\{{= \frac{{23\text{,}030} + {10\text{,}290\sqrt{5}}}{{2\text{,}481} + {392\sqrt{5}}}};}\end{matrix}$ $\begin{matrix}{c^{\prime} = \frac{\delta - \zeta}{{3\zeta} + \beta}} \\{= \frac{{{- 1}2\sqrt{5}} - {\sqrt{5}/7}}{{3\left( {\sqrt{5}/7} \right)} - \left( {\sqrt{5} + 7} \right)}} \\{{= \frac{85\sqrt{5}}{{4\sqrt{5}} + 49}};}\end{matrix}$ $\begin{matrix}{{{- 4}c^{\prime}} = {{- \left( \frac{340\sqrt{5}}{49 + {4\sqrt{5}}} \right)}\left( \frac{49 + {4\sqrt{5}}}{49 + {4\sqrt{5}}} \right)}} \\{{= {- \left( \frac{{6800} + {16\text{,}660\sqrt{5}}}{{2\text{,}481} + {392\sqrt{5}}} \right)}};}\end{matrix}$ ${{\begin{matrix}{{b^{2} - {4c^{\prime}}} = \frac{\left( {{23\text{,}030} + {10\text{,}290\sqrt{5}}} \right) - \left( {6800 + {16\text{,}660\sqrt{5}}} \right)}{\left( {49 + {4\sqrt{5}}} \right)^{2}}} \\{{= \frac{{16\text{,}230} - {6\text{,}370\sqrt{5}}}{\left( {49 + {4\sqrt{5}}} \right)^{2}}};}\end{matrix} \pm \sqrt{b^{2} - {4c^{\prime}}}} = \frac{\pm \sqrt{{16\text{,}230} - {6\text{,}370\sqrt{2}}}}{49 + {4\sqrt{5}}}};{and}$$\begin{matrix}{z_{R} = \frac{{- b^{\prime}} \pm \sqrt{b^{\prime^{2}} - {4c^{\prime}}}}{2}} \\{= \frac{{105} + {{49\sqrt{5}} \pm \sqrt{{16\text{,}230} - {6\text{,}370\sqrt{5}}}}}{{98} + {8\sqrt{5}}}} \\{= \frac{{105} + {{49\sqrt{5}} \pm \sqrt{\left( {{- 65} + {49\sqrt{5}}} \right)^{2}}}}{{98} + {8\sqrt{5}}}} \\{= \frac{{105} + {{49\sqrt{5}} \mp \left( {{65} - {49\sqrt{5}}} \right)}}{{98} + {8\sqrt{5}}}} \\{{= \frac{{40} + {98\sqrt{5}}}{{{98} + {8\sqrt{5}}}\;}};\frac{170}{98 + {8\sqrt{5}}}} \\{{= {\left( \frac{\sqrt{5}}{\sqrt{5}} \right)\left\lbrack \frac{{8\left( \sqrt{5} \right)^{2}} + {98\sqrt{5}}}{{98} + {8\sqrt{5}}} \right\rbrack}};\frac{170}{98 + {8\sqrt{5}}}} \\{{= \frac{\left. {{\sqrt{5}\left( {8\sqrt{5}} \right)} + 98} \right)}{{{98} + {8\sqrt{5}}}\;}};\frac{170}{98 + {8\sqrt{5}}}} \\{{= \sqrt{5}};{\frac{170}{{98} + {8\sqrt{5}}}.}}\end{matrix}$

Naturally, the last of such three famous continuous cubic equations, asstipulated above, alternatively could have been resolved algebraicallywithout having to resort to such cumbersome simultaneous reductionprocess.

This could be achieved simply by realizing that once a value of (becomesdesignated, an angle of 3θ magnitude that it is representative of veryeasily could be determined trigonometrically; whereby, a value for z_(R)which corresponds to its trisector, computed as being one-third of suchvalue, and thereby algebraically expressed merely as θ, thereafter alsocould be trigonometrically determined.

Unfortunately, the pitfall that accompanies such shortened algebraicprocess is that such common root, z_(R), does not become identifiedsolely by conventional Euclidean means!

The method to do so would be to draw straight lines whose lengths are ofmagnitudes which are equal to the value of roots belonging to suchabbreviated Quadratic Formula z_(R)=(½)(−b′±√{square root over(b′2−4c′)}), much in the same manner as was employed earlier whenquadratic roots first were determined by means of geometric constructionin FIG. 16.

For such algebraic determination, as made above, the magnitude of atrisector for an angle whose tangent is of a designated magnitude√{square root over (5)}/7 could be geometrically constructed by means ofapplying the following sequence of Euclidean operations; therebyrendering a particular pattern, as is depicted in FIG. 17:

two right triangles are drawn in the lower right corner which share acommon side of length (49+4√{square root over (5)})/100, and whose othermutual sides are of respective lengths:

(105+49√{square root over (5)})/100; and

85√{square root over (5)}/100;

such common side is extended to a unit length;

a perpendicular straight line is drawn above the newly formed endpointof such extension;

the hypotenuses appearing in such two previously drawn right trianglesare extended until they intersect such newly drawn perpendicularstraight line, thereby depicting two more similar right triangles;

whereby, the lengths of the unknown sides of such two newly drawn righttriangles can be determined by virtue of the proportions establishedbetween the known lengths of corresponding sides of their respectivesimilar right triangles and their common side of unit length, therebyenabling designations of −b′ length and c′ to be notated upon suchdrawing to reflect the following determinations:

$b^{\prime} = {{{- \left( \frac{{105} + {49\sqrt{5}}}{{4\sqrt{5}} + 49} \right)} - b^{\prime}} = {\left( \frac{{105} + {49\sqrt{5}}}{{4\sqrt{5}} + 49} \right)\left( \frac{{1/1}00}{{1/1}00} \right)}}$${\frac{- b^{\prime}}{1} = \frac{\left( {{105} + {49\sqrt{5}}} \right)/100}{\left( {49 + {4\sqrt{5}}} \right)/100}};$$c^{\prime} = {\frac{85\sqrt{5}}{{4\sqrt{5}} + 49}\left( \frac{1/100}{1/100} \right)}$${\frac{c^{\prime}}{1} = \frac{\left( {85\sqrt{5}} \right)/100}{{\left( {49 + {4\sqrt{5}}} \right)/1}00}};$

next, a square whose sides are of length −b′ is to be drawn, asindicated in the lower left-hand corner of FIG. 17;

a rectangle then becomes drawn whose base of unit length is to alignalong the lower side of such square and whose left lower corner is toshare the very position which the left hand lower corner of such squareoccupies;

a straight line then is to become drawn which extends from such newlyidentified common corner, passes through an intersection point which ismade between the upper side of such previously drawn square and theright side of such newly drawn rectangle, and thereafter continues as alarge diagonal until it intersects with the right side of suchpreviously drawn square;

the distance between such newly determined intersection point above thelower side of such square of base dimension −b′ is to become denoted asb′², as determined by the proportion established between thecorresponding sides of two new similar right triangles whose respectivehypotenuses align upon such just drawn long diagonal, whereby suchproportion becomes calculated as b′2/−b′=−b′/1;

a horizontal line next is set off a distance of b′² above the based ofsuch previously drawn square;

another horizontal line of is set off a distance of 4c′ above the basedof such previously drawn square;

the intervening length existing between them, amounting to a magnitudeof b′²⁻−4c′, must constitute the entire area of the small rectangle theyfurthermore describe, as bounded by the two opposite side of suchpreviously drawn rectangle whose base is equal to a length of unity;

a second square of unit base dimension then becomes described such thatits lower portion aligns directly upon such previously describedrectangle of area equal to b′²⁻−4c′;

a semicircle thereby can be drawn to the right of such square whosediameter aligns upon its left side;

straight lines thereafter are drawn from the respective ends of suchsemicircle diameter to the point residing upon its circumference whichintersects the horizontal straight line which resides at a distance ofb′² above the base of such previously drawn square whose respectivesides each are ‘b’ in length;

by virtue of the Pythagorean Theorem, such lower straight line, as drawnfrom the lower extremity of the diameter of such semicircle andextending to a point lying upon its circumference, must amount to alength which is equal to the square root of the b′²−4c′ area of suchpreviously described rectangle;

such length thereafter is reproduced as an extension to the horizontalstraight line previously drawn which resides a distance of 4c′ above thebase of the previously drawn square whose sides each equal −b′ inlength;

such new straight line extension is notated as being of overall length2z_(R)=−b′+√{square root over (b′²−4c′)}, as is indicated both at thevery the top and very bottom of such drawing; and

such overall length thereafter becomes bisected in order to distinguishand thereby designate a length z_(R) which amounts to one-half suchmagnitude.

Obviously, such geometric construction approach cannot pose a solutionfor the classical problem of the trisection of an angle; simply becausethe generalized cubic equation format that contributes to its verydetermination, specifically being z³−(√{square root over(5)}+7)z²+(7√{square root over (5)}+12)z−12√{square root over (5)}=0,could not be derived without a prior awareness of the very solutionitself.

A second less complicated example demonstrating that it is possible toapply algebraic information in order to create a geometric solution forthe problem of the trisection of an angle pertains to a generalizedcubic equation whose coefficients β and γ are set to zero, and whosecoefficient δ amounts to a value of +1, thereby establishing thespecific cubic equation z_(R) ³+1=0.

From such information, the following details can be gleaned:

z_(R)⁵ + 1 = 0 z_(R)⁵ = −1 $\begin{matrix}{z_{R} = \sqrt[3]{- 1}} \\{= {- 1}}\end{matrix}$ tan  θ = −1 $\begin{matrix}{\theta = {{arc}\mspace{14mu} \tan \mspace{14mu} \left( {- 1} \right)}} \\{{= {135{^\circ}}};}\end{matrix}$ $\begin{matrix}{{3\theta} = {3(\theta)}} \\{= {3\left( {135{^\circ}} \right)}} \\{{= {405{^\circ}}};{{and}\mspace{14mu} {as}\mspace{14mu} a\mspace{14mu} {check}}}\end{matrix}$ $\begin{matrix}{\zeta = {\left( {{3z_{R}} - z_{R}^{3}} \right)/\left( {1 - {3z_{R}^{2}}} \right)}} \\{= {\left( {{- 3} + 1} \right)/\left( {1 - 3} \right)}} \\{= {{- 2}/{- 2}}}\end{matrix}$ tan (3 θ) = +1 $\begin{matrix}{{3\theta} = {{arc}\mspace{14mu} \tan \mspace{14mu} \left( {+ 1} \right)}} \\{= {\left( {{360} + 45} \right){^\circ}}} \\{= {405{{^\circ}.}}}\end{matrix}$

Such algebraic determination, as made above, thereby enables thetrisection of an angle to be geometrically constructed as follows:

from a designated value of =tan (3θ)=+1, an angle designated as 3θ whichamounts to exactly 450 in magnitude first becomes geometricallyconstructed with respect to the +x-axis; and

from an algebraically determined common root value of z_(R)=−1, atrisecting angle designated as θ which amounts to exactly 135° inmagnitude thereafter becomes geometrically constructed with respect tothe +x-axis.

Needless to say, such geometric construction, as posed above, althoughrepresenting geometric solution for the problem of the trisection of anangle, nevertheless does not pose a solution for the classical problemof the trisection of an angle. This is because a value for such commonroot z_(R) cannot be ascertained solely by means of a geometricconstruction which proceeds exclusively from a given value of (=tan(3θ)=+1.

Although a straight line of slope z_(R)=−1 could be geometricallyconstructed rather easily from another line of given slope (=+1, suchgeometric construction pattern represents just one out of an infinitenumber of straight line possibilities which otherwise could bedistinguished geometrically from a given value of (=+1.

Hence, the sequence of Euclidean operations which governs suchtrisection can be completed with certainty only by incorporating suchalgebraic determination that z_(R)=+1, or else simply by algebraicallydividing such geometrically constructed 405 angle by a factor of three.

In either case, since both of such algebraic results are tied only tosuch 135° trisector of slope z_(R)=−1, the only way to determine suchinformation solely via straightedge and compass from a geometricallyconstructed 450 angle would be to distinguish them from the results of aEuclidean trisection which has not yet been performed.

Such process entails knowledge of the results of a geometricconstruction before it actually becomes conducted, thereby violating therules of conventional Euclidean practice which require that geometricconstruction can proceed only from a given set of previously definedgeometric data.

In order to further emphasize just how the use of aforehand knowledgeinadvertently creeps into conventional Euclidean practice, therebygrossly violating its very rules, a last rather telling example isafforded below whereby given angle NMP, as depicted in FIG. 1B, is to beof the very size which actually appears in such figure; thereby veryclosely amounting to twenty degrees. As such:

angle QPS, being geometrically constructed to three times that size,must be exactly sixty degrees. It becomes very easy to draw suchrendered angle because the internal angle of a geometrically constructedequilateral triangle is that same size;

the next step is to determine whether or not Euclidean operations can belaunched exclusively from such designated angle QPS in order to locatethe correct positions of points M and N; and

as it turns out, intersection points M and N cannot be distinguishedsolely via straightedge and compass solely from such rendered angle QPS.That is to say, there is absolutely no geometric construction that canbe performed with respect to such sixty degree angle QPS which canlocate points M and N, short of having aforehand awareness of theirrespective locations.

Such above analysis reveals that with respect to the particular geometryrepresented in such famous FIG. 1B Archimedes Euclidean formulation,when commencing only from angle QPS of designated sixty degreemagnitude, points M and N truly qualify as overlapment points.

Were this above assertion not to be true, it would be tantamount totrisecting such sixty degree angle QPS solely by means of applying astraightedge and compass to it; thereby solving the classical problem ofthe trisection of an angle without having any other predisposedknowledge and, in so doing, accomplishing a feat that is entirelyimpossible!

With regard to a prior discussion concerning the input box entitledPROBABILISTIC PROOF OF MATHEMATIC LIMITATION 10, it was mentioned thattrisection can be achieved by means of performing a multitude ofconsecutive angular bisections, all geometrically constructed upon justa single piece of paper.

Such approach generates a geometric construction pattern that isindicative of a geometric progression whose:

constant multiplier, “m”, is set equal to −½; and

first term, “f”, is algebraically denoted as 3θ.

Moreover, the overall sum, “s”, of such geometric progression consistingof an “n” number of terms can be reresented by the common knowledgeformula:

$\begin{matrix}{s = {{f\left( {m^{n} - 1} \right)}/\left( {m - 1} \right)}} \\{= {3\; {{\theta \left( {{{- 1}/2^{n}} - 1} \right)}/\left( {{{- 1}/2} - 1} \right)}}} \\{{= {{- 2}\; {\theta \left( {{{- 1}/2^{n}} - 1} \right)}}};}\end{matrix}\quad$

whereby

for an infinite number of terms, such equation thereby reduces to,

$\begin{matrix}{s = {{- 2}\; {\theta \left( {{{- 1}/2^{\infty}} - 1} \right)}}} \\{= {{- 2}\; {\theta \left( {0 - 1} \right)}}} \\{= {2\; {\theta.}}}\end{matrix}\quad$

Such result indicates that after conducting an infinite number ofsuccessive bisection operations, it becomes possible to geometricallyconstruct an angle that amounts to exactly ⅔ the size of an angle ofdesignated 3θ magnitude, whereby their difference then must distinguishits trisector.

Below, a method is furnished which describes how to geometricallyconstruct the first five terms appearing in such governing geometricprogression; and in so doing thereby assuming the form3θ−3θ/2+3θ/4−3θ/8+3θ/16=33θ/16.

In such development, the value of the first term, algebraically denotedas 3θ, can be set equal to virtually any designated magnitude that isintended to be trisected. By inspection, it furthermore becomes apparentthat the numerical value of each succeeding term is equal to one-halfthe magnitude of its predecessor. As such, values for such diminishingmagnitudes can be geometrically constructed merely by means of bisectingeach of such preceding angles.

Lastly, wherein positive values could applied in a counterclockwisedirection, negative magnitudes would appear in a completely opposite, orclockwise direction, with respect to them.

The specific details which pertain to a FIG. 18 drawing of this natureare itemized as follows:

an angle of magnitude 3θ is drawn such that its vertex aligns upon theorigin of an orthogonal coordinate system with its clockwise sideresiding along its +x-axis;

such given angle, being of magnitude 3θ, becomes bisected, whereby suchbisector resides at an angle relative to such +x-axis that amounts to½(3θ)=3θ/2;

the upper portion of such bisected angle, amounting to a size of 3θ/2,then itself becomes bisected, whereby a determination made as to thelocation of such second bisector would place it at an angle of3θ/2+3θ/4=9θ/4 with respect to the +x-axis;

the angle formed between such first bisector and second bisector nextbecomes bisected, whereby a determination made as to the location ofsuch third bisector would place it at an angle of 9θ/4−3θ/8=15θ/8 withrespect to the +x-axis; and

the angle formed between such second bisector and third bisector thenitself becomes bisected, whereby a determination made as to the locationof such fourth bisector would place it at an angle of 15θ/8+3θ/16=33θ/16with respect to the +x-axis.

Quite obviously, it remains possible to continue such activity untilsuch time that the naked eye no longer could detect a bisector for anarc that invariably becomes smaller and smaller with each subsequentbisection operation.

In this regard, the resolution of the naked eye is considered to belimited to about one minute of arc, thereby amounting to 1/60^(th) of adegree, whose decimal equivalent is 0.01667°.

Once the human eye no longer can detect gradations resulting from suchbisectors process, they could be located erroneously or evensuperimposed upon prior work.

Since the use of a microscope might increase such perceptioncapabilities, it might enable a few additional bisections to becomeaccurately determined. However, being that an infinite number ofbisections are needed in order to generate a precise trisector in thismanner, such enhancement only would serve to slightly improve upon theoverall approximation of any trisector which becomes produced.

The Successive Bisection Convergence Chart, as presented in FIG. 19,describes the results produced by such geometric progression as thenumber of terms is shown to increase in its first column, as headed bythe term n.

The second column therein is devoted to calculations which apply to suchgeometric progression, based upon the number of terms it contains. Ineach line item, the last value provided indicates the overall size ofthe angle which would become geometrically constructed by means ofconducting such successive bisection process.

Notice that FIG. 19 is discontinued at a value of n=22. This is because,at this point in such overall geometric construction process, anaccuracy of six decimal places, amounting to (2.000000)θ would becomerealized.

Since the only time that a bisection operation is not conducted is whenn=1, each successive line item within such FIG. 19 chart depicts ageometric construction pattern that could be generated by means ofperforming a total of n−1 bisection operations.

Hence, an accuracy of one-millionth could be obtained by means ofconducting twenty-one successive bisections.

The analysis presented below discloses that for a 20° trisector, suchabove summarized process of successive angular bisections would have tobe disbanded during the twelfth bisection operation due to the naked eyeno longer being able to discern the exact placement of its bisector.

As such, the number of terms this condition would apply to, as indicatedin such FIG. 19 chart, would be when n=13.

From such FIG. 19 chart, the separation needed to be distinguished whenperforming such twelfth bisection is calculated to be

$\begin{matrix}{{{2.000244\; \theta} - {1.999512\; \theta}} = {0.000732\theta}} \\{= {0.000732\left( {20{^\circ}} \right)}} \\{= {0.01464{{^\circ}.}}}\end{matrix}\quad$

Therefore, since such 0.01464° needed separation clearly is smaller thanthe 0.01667° which the naked eye is capable of perceiving; it means thatsuch twelfth bisector could be located erroneously.

When referring to FIG. 18, notice that an angle of size 3θ whose vertexis placed at the origin of a Cartesian Coordinate System such that itsclockwise side aligns upon its +x-axis is indicative of such geometricprogression for the particular condition when n=1.

Additionally, four subsequent bisections are depicted, each of which isconsidered to have been performed solely by conventional Euclideanmeans.

The purpose of the shading therein is to suitably distinguish betweeneach of such bisection activities as follows:

such angle of magnitude +3θ is bisected in order to distinguish twoseparate arcs, each being of 3θ/2 size;

with the upper portion of such bisected angle, amounting to a size of3θ/2, then itself becoming bisected, the determination made as to thelocation of such second bisector would place it at an angle of 3θ/4counterclockwise of such first bisector position;

with the angle formed between such first bisector and second bisector,amounting to a size of 3θ/4, then itself becoming bisected, thedetermination made as to the location of such third bisector would placeit at an angle of 3θ/8 clockwise of such second bisector position, addenoted by the minus sign notation; and

with the angle formed between such second bisector and third bisector,amounting to a size of 3θ/8, then itself becoming bisected, thedetermination made as to the location of such fourth bisector wouldplace it at an angle of 3θ/16 counterclockwise of such third bisectorposition.

As to the role which cube roots could play in a geometric solution ofthe problem of the trisection of an angle, below it is shown how todetermine the length of a straight line, half which amounts to its cuberoot value, whereby it could be algebraically stated that:

${\sqrt[3]{} = {/2}};$

such that by cubing both sides;

=

³/8

8

=

³

4(2)=

²

2√{square root over (2)}=

√{square root over (2)}=

/2; and

relevant information then is to be introduced in the form of an anglewhose complement furthermore turns out to be its trisector,algebraically determined as follows:

θ = 90^(∘) − 3 θ 3θ + θ = 90^(∘) 4 θ = 90^(∘) θ = 22.5^(∘)2 θ = 45^(∘) 3θ = 67.5^(∘);sin   (3θ) = 3  sin   θ − 4  sin³  θcos   (90 − 3 θ) = sin   θ  (3 − 4  sin²  θ) $\begin{matrix}{{\cos \mspace{14mu} \theta} = {\sin \mspace{14mu} {\theta \mspace{14mu}\left\lbrack {{(2)\left( {1 - {2\mspace{14mu} \sin^{2}\mspace{14mu} \theta}} \right)} + 1} \right\rbrack}}} \\{= {\sin \mspace{14mu} {\theta \mspace{14mu}\left\lbrack {{2\cos \mspace{14mu} \left( {2\; \theta} \right)} + 1} \right\rbrack}}} \\{= {\sin \mspace{14mu} \theta \mspace{14mu} \left( {{2\; \cos \mspace{14mu} 45{^\circ}} + 1} \right)}} \\{= {\sin \mspace{14mu} {\theta\left\lbrack {{(2)\left( \frac{\sqrt{2}}{2} \right)} + 1} \right\rbrack}}} \\{= {\sin \mspace{14mu} {\theta \left( {\sqrt{2} + 1} \right)}}}\end{matrix}$ $\frac{1}{\sqrt{2} + 1} = {\tan \mspace{14mu} \theta}$${\frac{1}{\sqrt{2} + 1}\left( \frac{\sqrt{2} - 1}{\sqrt{2} - 1} \right)} = {\tan \mspace{14mu} \theta}$$\frac{\sqrt{2} - 1}{2 - 1} = {\tan \mspace{14mu} \theta}$${\sqrt{2} - 1} = {\tan \mspace{14mu} \theta}$${\sqrt{2} - 1} = \frac{1}{\tan \left( {3\; \theta} \right)}$$\begin{matrix}{{\tan \left( {3\theta} \right)} = {\frac{1}{\sqrt{2} - 1}\left( \frac{\sqrt{2} + 1}{\sqrt{2} + 1} \right)}} \\{= \frac{\sqrt{2} + 1}{2 - 1}} \\{{= {\sqrt{2} + 1}};}\end{matrix}$

it therefore becomes possible to geometrically construct a righttriangle whose sides amount to respective lengths of 1 and 1+√{squareroot over (2)} such that its tangent, ζ, amounts to a value of1+√{square root over (2)};

whereby such √{square root over (2)} length is drawn as the hypotenuseof a 45θ right triangle, and such 1+√{square root over (2)} therebyrepresents the addition of its side added to such hypotenuse length; and

such hypotenuse of length √{square root over (2)} after becoming doubledand thereby amounting to 2√{square root over (2)}, being its cubedvalue, thereafter can be bisected in order to arrive at its cube root.

The algebraic cubic equation which correlates to this geometricconstruction process assumes the form of z_(R) ³+3z_(R)²+3z_(R)+(3−2ζ)=0; as determined below:

tan(3θ)=√{square root over (2)}+1=ζ

√{square root over (2)}=ζ−1; and

tan θ=z _(R)=√{square root over (2)}−1

z _(R)+1=√{square root over (2)}

(z _(R)+1)³=(√{square root over (2)})³

(z _(R)+1)³=2√{square root over (2)}

(z _(R)+1)³=2(ζ−1)

(z _(R) ³+3z _(R) ²+3z _(R)+1)−2(ζ−1)=0

z _(R) ³+3z _(R) ²+³ z _(R)+(3−2)=0.

To finalize a discussion raised earlier, FIG. 20 relates one complexnumber to another which serves both as its trisector, as well its cuberoot.

To elaborate upon this, complex numbers typically are representedgeometrically as straight lines which appear upon an xy plane known asthe complex plane.

Each straight line featured therein commences from the origin of arectilinear coordinate system, and contains an arrow at its terminationpoint to express direction.

The convention used to specify a complex number is first to indicate itsreal numerical magnitude, followed by its imaginary component. Suchimaginary aspect is represented by an Arabic letter, i, used to denotean imaginary term √{square root over (−1)}, followed by its magnitude.

As such, the coordinate values of complex number termination pointsdesignate their respective imaginary and real number magnitudes; therebyfully describing them.

In FIG. 20, such two complex numbers are shown to be expressed as cos(3θ)+i sin (3θ), and cos θ+i sin θ.

Conversely, since the ratio between the magnitudes of the real andimaginary portions of such first complex number is (sin 3θ)/(cos 3θ)=tan3θ, the straight line which represents it, by exhibiting such slope,thereby must pass through the origin while forming an angle of 3θ withsuch x-axis.

Likewise, the straight line which represents such second complex number,by exhibiting a slope of tan θ, thereby must pass through the originwhile instead forming an angle of θ with respect to the x-axis and, inso doing, trisecting such angle of 3θ magnitude.

The fact that the complex number cos θ±i sin θ also turns out to be thecube root of the first complex number cos (3θ)+i sin (3θ) furthermore isto be verified algebraically by applying the binomial expansion(A+B)³=A³+3A²B+3AB²+B³ for the express condition when the A=cos θ, andB=i sin θ as follows:

A³ + 3A²B + 3AB² + B³ = cos^(3  )θ + 3(cos²  θ)(i sin   θ) − 3(cos   θ)(sin²  θ) + (i sin   θ)³(A + B)³ = cos³  θ + 3(1 − sin²  θ)  (i sin   θ) − 3(cos   θ)(1 − cos²  θ) − i  sin³  θ     (cos   θ + i  sin   θ)³ = cos   (3θ) + i  sin   (3 θ)$\mspace{79mu} {{{\cos \; \theta} \pm {i\mspace{14mu} \sin \mspace{14mu} \theta}} = {\sqrt[3]{{\cos \mspace{14mu} \left( {3\theta} \right)} \pm {i\mspace{14mu} \sin \mspace{14mu} \left( {3\theta} \right)}}.}}$

Lastly, one final justification is about to be put forth, essentiallyclaiming that only an availability of overlapment points can fullyaccount for why the classical problem of the trisection of an anglecannot be solved!

Public sentiment on this topic, as highly influenced by the earlierdiscoveries of Wantzel and Galois dating all the back to the mid 1800's,instead generally leans to attributing an inability to geometricallyconstruct cube roots as being the principal cause which preventstrisection.

Moreover, at the very heart of this matter lies a fundamental issue ofconstructability.

To openly dispute such issue, upon drawing an angle of arbitrarilyselected magnitude, there is a good chance that its trigonometricproperties will turn out to be cubic irrational. This is because a fargreater number of angles exist which exhibit cubic irrationaltrigonometric properties than do other angles whose trigonometricproperties are of rational and quadratic irrational value.

From such initial angle, an entire geometric construction pattern couldbe generated which belongs to the Euclidean formulation, as posed inFIG. 13. Therein, such singular drawing would depict just how a givenangle VOO′ actually relates to rendered angle VOU′, amounting to exactlythree times its size, by virtue of specific trigonometric propertieswhich are inherent to each of such angles, as characterized by thefamous cubic equation sin (3θ)=3 sin θ−4 sin³ θ.

The basic problem with such scenario is that such drawing, althoughfully constructible by a process of sheer random selection, never couldbe repeated; thereby becoming relegated to approximation when attemptingto reproduce it.

More particularly stated, although the likelihood of drawing an anglewhich exhibits cubic irrational trigonometric properties is quite high,as due to a substantial availability of them, the probability ofgeometrically constructing a specific angle, even one which mightfeature a particular transcendental trigonometric property such a pi forexample, nevertheless approaches zero; being entirely consistent withthe previously stipulated premise that absolutely no cubic irrationallength can be geometrically constructed, but only approximated, from agiven unit length.

To further emphasize this outstanding difficulty, consider the largelyunknown fact that even the rarified transcendental number, π, can beapproximated by means of geometric construction well beyond what thenaked eye could detect.

To demonstrate this, a rational number very easily can be described bythe ratio of two cubic irrational numbers by an algebraic manipulationsuch as:

${\frac{13}{9} = {\frac{13}{9}\left( \frac{\tan \; 20{^\circ}}{\tan \; 20{^\circ}} \right)}};{whereby}$$\begin{matrix}{\frac{13}{9} = {\frac{13}{9}\frac{\tan \mspace{14mu} 20{^\circ}}{\tan \mspace{14mu} 20{^\circ}}}} \\{= \frac{4.7316130455\mspace{14mu} \ldots}{3.2757321084\mspace{14mu} \ldots}}\end{matrix}$

Similarly, the actual transcendental value of π can be multiplied to thesin 80° in order to produce another transcendental length as follows:

π sin 80°=3.093864802 . . . ; and

π(0.9848077530 . . . )=4(0.77346620052 . . . ).

Moreover, all of the stated values in such above equation, except for l,furthermore very closely could be approximated as actual rationalnumbers, down to a significance of at least ten decimal places; beingwell beyond the accuracy of what the naked eye could detect.

Such estimated result is furnished directly below, whereby allconstructible rational numbers thereby could be algebraically expressedas follows:

${{{\pi \left( \frac{984\text{,}807\text{,}753}{1\text{,}000\text{,}000\text{,}000} \right)} = {4\left( \frac{77\text{,}346\text{,}620\text{,}052}{100\text{,}000\text{,}000\text{,}000} \right)}}{{\pi \left( \frac{984\text{,}807\text{,}753}{1\text{,}000\text{,}000\text{,}000} \right)} = {4\left( \frac{19\text{,}336\text{,}655\text{,}013}{25\text{,}000\text{,}000\text{,}000} \right)}}{{\pi (L)} = {4(T)}}};{or}$π L = 4T.

Notice that such above described rational lengths 4, T, and L now can begeometrically constructed from an arbitrarily applied, or given lengthof unity.

In the above example, there is little need to attempt to reduce therational length T any further than is indicated. This is because it isnecessary only to know that a rational length ofT=19,336,655,013/25,000,000,000 could be made use of to geometricallyconstruct another length that very closely approximates the actual valueof pi.

From such equation πL=4T, as determined above, the proportion

$\frac{\pi}{T} = \frac{4}{L}$

readily could be established; whereby a very close estimation of thelength pi thereby could be identified from the geometric construction oftwo similar right triangles whose sides respectively consist of drawnrational lengths 4, T, and L. Understandably, the level of accuracyattributed would amount to only three, or perhaps four at the very most,significant digits.

To conclude, since transcendental lengths describe decimal sequenceswhich are considered to continue on indefinitely, they cannot be exactlygeometrically constructed from any long-hand division computation thatis indicative of a pair of rational numbers whose quotients begin torepeat themselves.

In the past, such difficulty merely was bypassed by means of consideringonly geometric construction patterns which could be redrawn.

Such process simply entails selecting a given angle whose trigonometricproperties are either rational or quadratic irrational. For example,upon considering a given angle VOO′ whose sine is equal to ⅓, thefollowing algebraic relationship could be obtained:

sin   θ = 1/3 θ = 19.47122063^(∘); and $\begin{matrix}{{\sin \mspace{14mu} \left( {3\theta} \right)} = {{3\; \sin \mspace{14mu} \theta} - {4\; \sin^{3}\mspace{14mu} \theta}}} \\{= {{3\left( {1/3} \right)} - {4\left( {1/3} \right)^{3}}}} \\{= {23/27}}\end{matrix}$ $\begin{matrix}{{3\theta} = {58{.4136619}{^\circ}}} \\{= {3{\left( {19{.47122063}{^\circ}} \right).}}}\end{matrix}$

Obviously the sin (3θ) also must be a rational value because it amountsto the sum of three times such selected rational value of ⅓ plus fourtimes the value of its cube; meaning that all coefficients within suchresulting equation 23/27=3 sin θ−4 sin³ θ very handily would consist ofonly rational numbers!

Accordingly, an associated geometric solution for the problem of thetrisection of an angle very easily could be drawn merely geometricallyconstructing an angle whose sine equals ⅓.

Notice, however, that such particular drawing would remain entirelyirreversible, despite being characterized by that very geometricconstruction pattern, as just described, belonging to the Euclideanformulation, as posed in FIG. 13; thereby specifically depicting a givenangle VOO′ which would exhibit a sine value of exactly ⅓. In otherwords, the claim that the classical problem of the trisection of anangle cannot be solved becomes further bolstered, even for an anglewhose sine value amounts to 23/27; as predicated upon the fact that anavailability of overlapment points must remain at work which preventssuch drawing from fully being backtracked upon. Naturally, in suchspecific case, relevant data, as previously stipulating that the sine ofthe trisector for such angle would amount to exactly ⅓, only wouldqualify as extraneous information, whereby its use would violate thevery Euclidean requirements which just so happen to be levied upon suchproblem.

Next, the issue of attempting to extract cube roots is to be addressed.In order to do this, consider that some Euclidean formulation somedaymight become devised, each of whose constituent geometric constructionpatterns would be fully reversible, as well as exhibit a rendered lengththat amounts to the cube of its given length. In so doing, it naturallywould follow that for each of such singular drawings, a cube root ofsuch rendered length value thereby could be geometrically constructedwithout having to introduce any additional relevant information.

Now, if a Euclidean formulation of such nature truly could be devised,an overriding question then would be whether such capability could insome way overcome the irreversible nature of any geometric constructionpattern in which the magnitude of a rendered angle amounts to exactlythree times the size of its given angle. For instance, could suchmagical Euclidean cube root capability enable angle VOU′, as appearingupon the irreversible representative geometric construction pattern forsuch Euclidean formulation, as posed in FIG. 13, to be fully backtrackedupon all the way to given angle VOO′ in order to solve the classicalproblem of the trisection of an angle?

Naturally, an activity of this nature would be severely limited in thatsome far-fetched reversible Euclidean cube root capability only could beapplied to any known aspect of such rendered angle VOU′. Such is thecase because when attempting to solve the classical problem of thetrisection of an angle, other lengths in FIG. 13, such as sin³ θ, stillwould remain unknown. Since it is impossible to take the cube root of anunknown value, such very difficulty would thwart any attempts to fullybacktrack from rendered angle VOU′ all the way back to given angle VOO′.

Accordingly, it is conjectured that some as yet undeveloped Euclideancapability to extract cube roots would have little to no impactwhatsoever upon enabling the classical problem of the trisection of anangle to become solved; as based upon the fact that such hypotheticalcube root development couldn't possibly offset the irreversibility ofsuch FIG. 13 representative geometric construction pattern. Oncerecognizing that it otherwise must be an availability of overlapmentpoints which actually prevents a backtracking activity of this naturefrom being accomplished, it becomes rather obvious that an introductionof any professed Euclidean cube root capability couldn't possiblyrectify a plaguing Euclidean irreversibility limitation which insteadactually prevents the classical problem of the trisection of an anglefrom actually being solved!

In closing, it is important to note that vital input leading to the verydiscovery of significant findings, as presented herein, never even wouldhave been obtained had it not been for one strange incident whichoccurred in 1962. It was then, that my high school geometry teacherinformed me that it was impossible to perform trisection solely byconventional Euclidean means. Her disclosure moved me greatly. I becomeintrigued; thereby fueled with a relentless curiosity to ascertainsecrets needed to unlock a trisection mystery that had managed to bafflemathematicians for millennia!

Naturally, during such prolonged fifty-five year investigation, certaincritical aspects pertaining to trisection became evident well ahead ofothers. For example, I realized that a general perception of geometrydating back all the way to the time of Archimedes perhaps might bebetter served by means of now considering a much needed extension to it;one that would transcend beyond the confines of conventional Euclideanpractice, and amplify even upon Webster's own definition of such word;whereby from an availability of straight lines, intersection points,circles, triangles, rectangles and parallelograms, leading to an overallprofusion of spheres, prisms and even pyramids, eventually would emergethe far greater understanding that any visualization which could bemathematically interpreted diagrammatically should be considered to beof a geometric nature!

Such enhanced perception would apply to real world events whereincertain articulating mechanisms, even those capable of performingtrisection, would be credited for accomplishing specific geometric featsthat otherwise could not be matched solely by conventional Euclideanmeans. Certain famous convolutions then would comprise known geometricshapes, such as the Conchoid of Nicomedes, the Trisectrix of Maclaurin,the catenary or hyperbolic cosine, the elliptical cone, the parabola,the Folium of Decartes, the Limacon of Pascal, the Spiral of Archimedes,the hyperbolic paraboloid, as well as logarithmic and even exponentialcurves; as previously were considered to be taboo within an otherwiselimited realm of conventional Euclidean practice.

Revolutionary material, as presented herein, consists largely of awealth of information that can be traced directly to a newly establishedmethodology that, in turn, is predicated upon a proposed extension toconventional Euclidean practice. In order to succeed at developing suchrather unconventional output, it became essential to take good notesover extended periods of time. Moreover, copyrights conveniently servedto document dates pertaining to significant discoveries.

Many concepts, as expressed herein, stem from a far broader pretextwhich previously was referred to as equation sub-element theory Uponreading my unfinished treatise entitled, The Principles of EquationSub-element Theory; United States Copyright Number TXu 1-960-826 grantedin April of 2015, it would become apparent that such purported new fieldof mathematics unfortunately only is in its embryonic stage ofdevelopment. By no means should it be considered to be complete! Infact, such document already was amended under United States CopyrightNumber TXu 1-976-071 during August of 2015, and presently is undergoingyet another revision in order to keep abreast with recent findings, someof which are to be disseminated to the public for the very first timeherein. Such copyright process permits premature theories to becomedocumented, and thereafter revised without difficulty in order tosuitably become refined into viable output.

Any prior art issue which might arise concerning the concurrentpreparation of two documents which might contain somewhat similar, oreven closely related information could be reconciled by means ofcontrolling which becomes published and/or disseminated first.

In this regard, such above described copyrighted material should pose noproblem because it never before was published, nor even disseminated tothe general public in any manner whatsoever. Hence, there is nocompelling reason to suspect that information contained therein mightqualify as prior art material. Such position is predicated upon onebasic understanding; being, that because the exclusive right granted bysuch copyrights to reproduce and/or distribute never before wasexercised, it becomes impossible for anyone to be aware of the verynature of such material.

Conversely, if the argument that such copyrighted material actuallyshould qualify as prior art otherwise were to persist in some thoroughlyunabated manner, it then would require a review by some expert who, bygaining access in some surreptitious manner to undisclosed information,thereby independently only would collaborate that such unfinishedcopyrighted information is seriously flawed. For example, suchhypothetical review would reveal that the term transcendental was usedinappropriately throughout such copyright and amendment thereto. Todaysuch mistake can be easily explained by mentioning that a thoroughunderstanding of Al-Mahani's work was gained only after such copyrightedinformation first became amended. Therefore, the correct replacementterm, being cubic irrational, couldn't possibly have appeared in earlierforms of such copyrights. Moreover, had such copyrighted informationbeen released to the public, well before it completion, then inaccurateinformation stating that only transcendental values, as consisting of alimited subset of all cubic irrational numbers, could be automaticallyportrayed by means of performing trisection; thereby contradictingcorrect details as presented herein.

Regarding the 2½ year interim which elapsed between the granting of suchtwo 2015 copyrights and the present day completion of this disclosure,such period of time is indicative of an expected turnaround needed toeffectively update information that well should be construed to includecomplex revolutionary material, thereby exceeding that of evolutionaryprojects by some considerable degree; whereby more leniency should beextended for their proper update.

By means of documenting what might appear to be similar theoryconcurrently in dual records, a process of leap frog would unfold,whereby what might have seemed to be credible information appearing in acopyrighted document, when worked upon earlier, soon would becomeoutdated by a subsequent accounting, such as this one; therebynecessitating yet another revision of such copyrighted document to becompleted before its release in order to remain totally consistent withrefinements now incorporated herein.

Accordingly, by means of publishing the contents of this disclosure wellahead of any portion of such, as yet undisclosed 500+ page copyrightedtreatise, this document shall be the first to become disseminatedanywhere on earth. Lastly, whereas such copyrights, as identifieddirectly above, evidently do not appear to qualify as prior art, itthereby should not be necessary to furnish a copy of them along with thesubmittal of this patent disclosure.

What is claimed is:
 1. A Euclidean formulation, being a practical meansfor representing upon just a single piece of paper an entire family ofgeometric construction patterns which can be exclusively derived from aspecific sequence of Euclidean operations; wherein only a double arrownotation would need to be placed at some strategic location upon one ofsuch geometric construction patterns, furthermore referred to as itsrepresentative geometric construction pattern, thereby making itpossible to observe how the appearance of such drawing would changethere due to modifications made to the size of an angle which appearselsewhere upon such drawing that becomes denoted algebraically by theGreek letter θ.
 2. A geometric forming process which theorizes thatcomplete pathways which lead from device settings of an angle trisectorall the way back to their corresponding trisectors furthermore could beshown to superimpose upon respective geometric construction patternsbelonging to a specific Euclidean formulation; thereby validating: thatsuch mechanism actually could portray a wide range of unique motionrelated solutions for the problem of the trisection of an angle; thatthe Euclidean limitation of being unable to backtrack upon irreversiblegeometric construction patterns whose rendered angles amount to exactlythree times the size of their respective given angles can be overcome bymeans of introducing motion; and that the reason why the classicalproblem of the trisection of an angle cannot be solved is because anydevised geometric construction pattern remains impervious to the effectsof time.
 3. A mathematics demarcation whose geometric forming processportion, being that wherein applied motions can affect outcomes, aloneis capable of portraying lengths which are of cubic irrational value; asopposed to its conventional geometric construction portion whereinfinite lengths of such magnitudes instead only could be approximated bymeans of applying a straightedge to a given length of unity.